View source for 600-cell

{{Short description|Four-dimensional analog of the icosahedron}}
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{{Infobox polychoron |
  Name=600-cell|
  Image_File=Schlegel_wireframe_600-cell_vertex-centered.png|
  Image_Caption=[[Schlegel diagram]], vertex-centered<br>(vertices and edges)|
  Type=[[Convex regular 4-polytope]]|
  Last=[[Rectified 600-cell|34]]|
  Index=35|
  Next=[[Truncated 120-cell|36]]|
  Schläfli={3,3,5}|
  CD={{CDD|node_1|3|node|3|node|5|node}}|
  Cell_List=600 ([[Tetrahedron|{3,3}]]) [[Image:Tetrahedron.png|20px]]|
  Face_List=1200 [[triangle|{3}]]|
  Edge_Count=720|
  Vertex_Count= 120|
  Petrie_Polygon=[[Triacontagon#Petrie polygons|30-gon]]|
  Coxeter_Group=H<sub>4</sub>, [3,3,5], order 14400|
  Vertex_Figure=[[Image:600-cell verf.svg|80px]]<br>[[icosahedron]]|
  Dual=[[120-cell]]|
  Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
}}
[[File:600-cell net.png|thumb|right|[[Net (polyhedron)|Net]]]]
In [[geometry]], the '''600-cell''' is the [[convex regular 4-polytope]] (four-dimensional analogue of a [[Platonic solid]]) with [[Schläfli symbol]] {3,3,5}.
It is also known as the '''C<sub>600</sub>''', '''hexacosichoron'''<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hexacosihedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref>
It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and a '''[[polytetrahedron]]''', being bounded by tetrahedral [[Cell (geometry)|cells]].

The 600-cell's boundary is composed of 600 [[Tetrahedron|tetrahedral]] [[Cell (mathematics)|cells]] with 20 meeting at each vertex.
Together they form 1200 triangular faces, 720 edges, and 120 vertices.
It is the 4-[[Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[icosahedron]], since it has five [[Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[triangle]]s meeting at every vertex.
Its [[dual polytope]] is the [[120-cell]].

== Geometry ==

The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration, the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units.
They must be algebraically converted to compare polytopes of unit radius.}} within the same radius.
The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest.
Complexity (as measured by comparing [[#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering.
This provides an alternative numerical naming scheme for regular polytopes in which the 600-cell is the 120-point 4-polytope: fifth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.|name=polytopes ordered by size and complexity|group=}}
It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}}

The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.
The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), "600-cell" column <sub>0</sub>''R/l'' {{=}} 2𝝓/2}} which is the [[golden ratio]].

{{Regular convex 4-polytopes}}

=== Coordinates ===

==== Unit radius Cartesian coordinates ====
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length {{sfrac|1|φ}} ≈ 0.618 (where φ = {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows:

8 vertices obtained from

:(0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

:(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})

The remaining 96 vertices are obtained by taking [[even permutation]]s of

:(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ<sup>−1</sup>|2}}, 0)

Note that the first 8 are the vertices of a [[16-cell]], the second 16 are the vertices of a [[tesseract]], and those 24 vertices together are the vertices of a [[24-cell]].
The remaining 96 vertices are the vertices of a [[snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}

When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[icosian]]s.

In the 24-cell, there are [[24-cell#Squares|squares]], [[24-cell#Hexagons|hexagons]] and [[24-cell#Triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}}
In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint.
In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}}
In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}
Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]].

The 60 axes and 75 16-cells of the 600-cell constitute a [[Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60<sub>5</sub>75<sub>4</sub> to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}}
Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.

==== Hopf spherical coordinates ====
In the 600-cell there are also great circle [[pentagon]]s and [[decagon]]s (in central planes through ten vertices).{{Sfn|Denney|Hooker|Johnson|Robinson|2020}}

Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell.
Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons.
Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}}
Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}}
Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon.
The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}}

By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[completely orthogonal]] squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex.
This latter pentagonal symmetry of the 600-cell is captured by the set of [[Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>){{Efn|name=Hopf coordinates|The [[Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles:

: (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>)

that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.
A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉<sub>''i''</sub> and 𝜉<sub>''j''</sub> are angles of rotation in the two [[completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]].
The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉<sub>''i''</sub>, 0, 𝜉<sub>''j''</sub>) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude").
The (𝜉<sub>''i''</sub>, {{sfrac|𝜋|2}}, 𝜉<sub>''j''</sub>) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles.
The other Hopf coordinates (𝜉<sub>''i''</sub>, 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉<sub>''j''</sub>) describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}}
Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>) to unit-radius Cartesian coordinates (w, x, y, z) is:<br>
: w {{=}} cos 𝜉<sub>''i''</sub> sin 𝜂
: x {{=}} cos 𝜉<sub>''j''</sub> cos 𝜂
: y {{=}} sin 𝜉<sub>''j''</sub> cos 𝜂
: z {{=}} sin 𝜉<sub>''i''</sub> sin 𝜂
The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as:

: ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}})

where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5).
The 𝜉<sub>''i''</sub> and 𝜉<sub>''j''</sub> coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell.
These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}}

=== Structure ===

==== Polyhedral sections ====
The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋.
Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏<sup>−1</sup>) beginning with a vertex}}{{Sfn|Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout|Parker|1998}}
These can be seen in the H3 [[Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}

:[[File:600-cell-polyhedral levels.png|640px]]

These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid).
Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane).
In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center.
But its own center is in the interior of the 600-cell, not on its surface.
V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell.
Thus V is the apex of a [[Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron.

{| class=wikitable
!colspan=2|Concentric Hulls
|-
|align=center|[[Image:Hulls of H4only-orthonormal.png|360px]]
|The 600-cell is projected to 3D using an orthonormal basis.
The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br>
<br>
1) two points at the origin<br>
2) two icosahedra<br>
3) two dodecahedra<br>
4) two larger icosahedra<br>
5) and a single icosidodecahedron<br>
<br>
for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices.
|-
|}

==== Golden chords ====
[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]]
The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]].
The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc.
The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
{{see also|24-cell#Hypercubic chords}}
The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏<sup>−1</sup>) beginning with a vertex; see column ''a''}} at eight different [[Chord (geometry)|chord]] lengths from each other.
These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}}
In ascending order of length, they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=The fractional square roots are given as decimal fractions where:
{{indent|7}}𝚽 ≈ 0.618 is the inverse golden ratio <math>\tfrac{1}{\phi} = \phi^{-1}</math>
{{indent|7}}𝚫 = 1 - 𝚽 = 𝚽<sup>2</sup> ≈ 0.382<br>
For example:
{{indent|7}}𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618|name=fractional square roots|group=}}

Notice that the four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons.
The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[golden ratio]] <big>φ</big> {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to <big>𝜋</big>:<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=John Carlos Baez|access-date=10 October 2022}}</ref><br>
: {{sfrac|𝜋|5}} = arccos ({{sfrac|φ|2}})
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing <big>φ</big> as a function of <big>𝜋</big> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br>
: <big>φ</big> = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the <big>φ</big> = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is <big>φ</big>, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the <big>φ</big> chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <big>φ</big>{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

==== Boundary envelopes ====
[[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]].
The 3D surface made of 600 tetrahedra is visible.]]
The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell.
The other 96 vertices constitute a [[snub 24-cell]].
Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}}
The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells.
One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: tetrahedra of edge length {{sfrac|1|φ}} ≈ 0.618 instead of octahedra of edge length 1.
It encloses the {{radic|1}} edges of the 24-cells, which become invisible interior chords in the 600-cell, like the [[24-cell#Hypercubic chords|{{radic|2}} and {{radic|3}} chords]].

[[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell]] performing a [[24-cell#Simple rotations|simple rotation]].
The 3D surface made of 24 octahedra is visible.
It is also present in the 600-cell, but as an invisible interior boundary envelope.]]
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of {{sfrac|1|φ}}, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[Tesseract#Radial equilateral symmetry|radially equilateral]].
Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[equilateral triangle]]s which all meet at the center.{{Efn|The long radius (center to vertex) of the [[24-cell]] is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[Golden triangle (mathematics)|golden triangle]]s which all meet at the center. All the [[regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[right triangle]]s which meet at the center.{{Efn|The [[orthoscheme]] is the generalization of the [[right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[golden ratio]] to its edge length; thus its radius is <big>φ</big> if its edge length is 1, and its edge length is {{sfrac|1|φ}} if its radius is 1.}}
Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[icosidodecahedron]], and the two-dimensional [[Decagon#The golden ratio in decagon|decagon]].
(The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.)
'''Radially golden''' polytopes are those which can be constructed, with their radii, from [[Golden triangle (mathematics)|golden triangles]].

The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes).
The shape of those interstices must be an [[Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}}
Therefore the successor may be constructed by placing [[Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor.
Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[cubic pyramid]]s.
But if we place 24 canonical [[octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell.
Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}}

==== Geodesics ====
The vertex chords of the 600-cell are arranged in [[geodesic]] [[great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§4 The planes of the 600-cell|pp=437-439}}

[[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[stereographic projection]] of the 600-cell's 72 central decagons onto their great circles.
Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]]
The {{radic|0.𝚫}} = 𝚽 edges form 72 flat regular central [[decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}}
Just as the [[icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10).
The 720 {{radic|0.𝚫}} edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, {{radic|3.𝚽}} apart.
As in the decagon and the icosidodecahedron, the edges occur in [[Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope.
The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[Clifford parallel]] great circles spanned by a twisted [[Annulus (mathematics)|annulus]].]][[Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[Hopf fibration|Hopf fiber bundles]] which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}

The {{radic|1}} chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|1}} chords join vertices which are two {{radic|0.𝚫}} edges apart.
Each {{radic|1}} chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[triangular bipyramid]]), and passes through the center of the shared face.
As there are 1200 faces, there are 1200 {{radic|1}} chords, in 600 parallel pairs, {{radic|3}} apart.
The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell|Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively.
The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs.
Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}}
The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}

The {{radic|1.𝚫}} chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}}
The {{radic|1.𝚫}} chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon.
The {{radic|1.𝚫}} chords join vertices which are two {{radic|0.𝚫}} edges apart on a geodesic great circle.
The 720 {{radic|1.𝚫}} chords occur in 360 parallel pairs, {{radic|2.𝚽}} = φ apart.

The {{radic|2}} chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex).
The {{radic|2}} chords join vertices which are three {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart).
There are 600 {{radic|2}} chords, in 300 parallel pairs, {{radic|2}} apart.
The 450 great squares (225 [[completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}

The {{radic|2.𝚽}} = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is of length {{radic|3.𝚽}}.
The {{radic|2.𝚽}} chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|2.𝚽}} chords, in 360 parallel pairs, {{radic|1.𝚫}} apart.

The {{radic|3}} chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells).
Each set of 32 triangles consists of the 96 {{radic|3}} chords and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|3}} chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The {{radic|3}} chords join vertices which are four {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart on a geodesic great circle).
Each {{radic|3}} chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 {{radic|1}} cubic cells.
The 1200 {{radic|3}} chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each {{radic|3}} chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}}
There are 1200 {{radic|3}} chords, in 600 parallel pairs, {{radic|1}} apart.

The {{radic|3.𝚽}} chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex.
The third edge (base) of each isosceles triangle is an edge of the pentagon of length {{radic|1.𝚫}}, so these are [[Golden triangle (mathematics)|golden triangles]]. The {{radic|3.𝚽}} chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|3.𝚽}} chords, in 360 parallel pairs, {{radic|0.𝚫}} apart.

The {{radic|4}} chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell.
The {{radic|4}} chords join opposite vertices which are five {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells.

The sum of the squared lengths{{Efn|The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.}} of all these distinct chords of the 600-cell is 14,400 = 120<sup>2</sup>.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}}
These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[completely orthogonal]] to a great 30-gon{{Efn|A ''[[triacontagon]]'' or 30-gon  is a thirty-sided polygon.
The triacontagon is the largest regular polygon whose interior angle is the sum of the [[interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[equilateral triangle]] (60°) and the [[regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell.
The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly.
The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords.
It does not touch any edges or vertices, but it does hit faces.
It is the central axis of a spiral skew 30-gram, the [[Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}}
Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all.
There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}}

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells).
These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[icosidodecahedron]].
There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes.
(More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}}
Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space.
Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens|Seidel|1973}}
Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together.
Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration.
Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles.
Vectors in 4-space may be combined by [[Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[William Rowan Hamilton|Hamilton]].{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0).
Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector.
Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br>
<small><math display=block>\begin{pmatrix}
w_2\\
x_2\\
y_2\\
z_2
\end{pmatrix}
*
\begin{pmatrix}
w_1\\
x_1\\
y_1\\
z_1
\end{pmatrix}
=
\begin{pmatrix}
{w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\
{w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\
{w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\
{w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1}
\end{pmatrix}
</math></small>}}
The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}}
Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}}
Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell.
Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}}
Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}}
Planes which are isoclinic have [[Clifford parallel]] great circles.{{Efn|name=Clifford parallels}}
A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}}
Each great square plane is [[completely orthogonal]] to another great square plane.
Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one {{radic|4}} long diameter): a great [[digon]] plane.{{Efn|In the 24-cell each great square plane is [[completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[digon]] plane.|name=digon planes}}
Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane.
Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere.
Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere.
Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell.
In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}}

==== Fibrations of great circle polygons ====
Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}}
Each [[fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S<sub>3</sub>|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}}
The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets.
The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell|Semple|1971|loc=§4. Isoclinic planes in Euclidean space E<sub>4</sub>|pp=6-7}}

===== Decagons =====
[[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six]] configuration 30<sub>2</sub>12<sub>5</sub> characteristic of the H<sub>4</sub> polytopes.{{^|Efn|name=Schläfli double six}} The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]]
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.

Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}}
Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations.
The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}}

The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[Hopf fibration]]s make the 600-cell a [[Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30<sub>2</sub>12<sub>5</sub>.{{^|Efn|name=Schläfli double six}}
It is called the [[Schläfli double six]] configuration after [[Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3<sup>2, 2, 1</sup>] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}}

===== Hexagons =====
The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons.
Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon.
Each octahedral cell occupies only one cell ring in each of the 4 fibrations.
The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration.

The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}
It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. 
The Hopf map of this fibration is the [[dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}
Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations.
The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.

===== Squares =====
The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares.
Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each.
Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations.
The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration.

The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells.
It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the {{radic|2}} tetrahedral cells of the 75 inscribed 16-cells, ''not'' the {{radic|0.𝚫}} tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} 
The Hopf map of this fibration is the [[icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}
Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations.

===== Clifford parallel cell rings =====
The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, studied cell rings in the general case of their geometry and [[group theory]], identifying each cell ring as a [[polytope]] in its own right which fills a three-dimensional manifold (such as the [[3-sphere]]) with its corresponding [[Honeycomb (geometry)|honeycomb]].{{Efn|name=orthoscheme ring}}
He found that cell rings follow [[Petrie polygon]]s and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms.
Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}}
Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[Clifford torus]], showed how the honeycombs correspond to [[Hopf fibration]]s, and made decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S<sub>3</sub>|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces.
Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}}
The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells.
They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}}
The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell).
The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell.
The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells.
In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}}
This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}}
The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings.
Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration).
In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}}
The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration.
All the fibrations have this two level structure with ''subfibrations''.

The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon.
Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.

The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square.
Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}}

The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope.
Each fibration is the entire 4-polytope.}}
The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}}
On this map each great circle fiber appears as a single point.
The base of a great decagon fibration of the 600-cell is the [[icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}
To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration.
But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell.
Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell.
Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Sfn|Sadoc|Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R<sup>3</sup>. It would not occur if the molecules were embedded in the non-Euclidean space of the [[3-sphere]] S<sup>3</sup>, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,{{Efn|name=Clifford parallels}} along which the molecules can be aligned without any conflict between compactness and [[torsion of a curve|torsion]].... The fibres of this [[Hopf fibration]] are great circles of S<sup>3</sup>, the whole family of which is also called the [[Clifford parallel]]s. Two of these fibers are C<sub>∞</sub> symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S<sup>3</sup>.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint."}}|name=Hopf fibration base}} of these fibrations on the 2-sphere).
Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 red faces of the [[Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration'').
The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}}

=== Constructions ===
The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}}
Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial.
The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.

==== Gosset's construction ====
[[Thorold Gosset]] discovered the [[Semiregular polytope|semiregular 4-polytopes]], including the [[snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius.
Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form.
In the first, more complex step (described [[Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}
In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}}

The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}}
The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells.
The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.

Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps.
The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation.
Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[midsphere]] to construct its outer [[Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual.
That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.

==== Cell clusters ====
Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]],{{Efn|name=tetrahedral cell adjacency}} or how they lie on the underlying surface envelope of the 24-cell's octahedral cells.
For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.

Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences....
[Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains.
Coxeter made full use of it, and expected the reader to use it....
Visualization is one of the most powerful interiorization techniques.
It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module.
Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional space that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}}
In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

===== Icosahedra =====
[[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[Regular icosahedron#Symmetries|snub octahedron]].
This term refers specifically to a [[Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}}
Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces.
The apex of the [[icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible).
The four cells lie in different hyperplanes.|alt=|200x200px]]
The [[vertex figure]] of the 600-cell is the [[icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center.
Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there.
However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices.
Thus the vertex icosahedron is actually a canonical [[icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|The icosahedron is not radially equilateral in Euclidean 3-space, but an icosahedral pyramid is radially equilateral in the curved 3-space of the 600-cell's surface (the [[3-sphere]]). In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the [[icosahedral pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that curved 3-space''. In Euclidean 4-space 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=vertex icosahedral pyramid|group=}}
Twenty tetrahedral cells meet at each vertex, forming an [[icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron.
The 600-cell has a [[dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}}

An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra.
Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five).
Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells.
Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}}

The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell.
The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[snub 24-cell]], which has exactly the same [[Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing.

The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces.
Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces.
Each pentagon has five distinct rotational orientations.
Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}}
Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell.
Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids.
Their apexes are the vertices of a [[24-cell#Hexagons|great circle hexagon]].
This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[triangular bipyramid]]) is a hexagon edge (a 24-cell edge).
There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a {{radic|1}} octahedral cell, but in the larger {{radic|2}} octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron.
The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]].
The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}}
The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}}
The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}}
The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron;{{Sfn|Itoh|Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[Jitterbug transformation|Jitterbug]] by [[Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}}
The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca|Al-Mukhaini|Koca|Al Qanobi|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[3-sphere|S<sup>3</sup>]] is similar to the [[kinematics of the cuboctahedron]] and the [[Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[Three-dimensional space|R<sup>3</sup>]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[Buckminster Fuller]].<ref>{{cite journal
 | last = Verheyen | first = H. F.
 | doi = 10.1016/0898-1221(89)90160-0
 | issue = 1–3
 | journal = [[Computers and Mathematics with Applications]]
 | mr = 0994201
 | pages = 203–250
 | title = The complete set of Jitterbug transformers and the analysis of their motion
 | volume = 17
 | year = 1989| doi-access = free
 }}</ref>}}

The tetrahedral cells are face-bonded into [[Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[Boerdijk-Coxeter helix]].
This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]].
In the 600-cell we find them bent in the fourth dimension into geodesic rings.
Each ring has 30 cells and touches 30 vertices.
The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}}
5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge).
A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[Hopf fibration]].
There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}}
The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other.
Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex.
Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[Skew lines|skew edges]] of the tetrahedron.
Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron.
However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[Hopf fibration|Hopf fiber bundle]] of 12.
Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]].

The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same.
One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}}
Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}}

===== Octahedra =====
There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell.

Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells.
The central cell is the first section of the 600-cell beginning with a cell.
By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.

First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length {{radic|1}}.
Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs.
They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length {{radic|1}}.
They form a tetrahedron of edge length {{radic|1}}, which is the second section of the 600-cell beginning with a cell.{{Efn|The {{radic|1}} tetrahedron has a volume of 9 {{radic|0.𝚫}} tetrahedral cells.
In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it.
The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the {{radic|1}} tetrahedron.
The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}}
There are 600 of these {{radic|1}} tetrahedral sections in the 600-cell.

With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster.
The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length {{radic|1}}, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}}
As partially filled so far (by 17 tetrahedral cells), this {{radic|1}} octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each {{radic|1}} edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells).
In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces.
Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}}
Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A {{radic|1}} octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell.
The same {{radic|1}} octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point {{radic|1}} octahedral section, a 4-point {{radic|1}} tetrahedral section, and a 4-point {{radic|0.𝚫}} tetrahedral section.
In the curved three-dimensional space of the 600-cell's surface, the {{radic|1}} octahedron surrounds the {{radic|1}} tetrahedron which surrounds the {{radic|0.𝚫}} tetrahedron, as three concentric hulls.
This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}}

Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical {{radic|1}} [[octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter {{radic|0.𝚫}} edges, replacing the apex with four vertices.
The truncation has also created another four vertices (arranged as a {{radic|1}} tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with {{radic|0.𝚫}} edges.
The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all.
The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two {{radic|1}} edges (and just one of those routes ran through the single apex).
The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three {{radic|0.𝚫}} edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length {{sfrac|1|φ}} ≈ 0.618.

===== Union of two tori =====
There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell.
An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines".

[[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}}
Its opposite edges are identified, forming a [[duocylinder]].]]
The [[120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]].
Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex.
The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}}

Start by assembling five tetrahedra around a common edge.
This structure looks somewhat like an angular "flying saucer".
Stack ten of these, vertex to vertex, "pancake" style.
Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron.
You can view this as five vertex-stacked [[icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet.
This 10-cell ring is shaped like a [[pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center.
This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}}
Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}}
The surface is the same as that of ten stacked [[pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}}
Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices.
Stack another tetrahedron on each exposed face.
This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge.
The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}}
The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons).
These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes).

Build a second identical torus of 250 cells that interlinks with the first.
This accounts for 500 cells.
These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges.
This latter set of 100 tetrahedra are on the exact boundary of the [[duocylinder]] and form a [[Clifford torus]].{{Efn|A [[Clifford torus]] is the [[Hopf fibration|Hopf fiber bundle]] of a distinct [[SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[3-sphere]], involving all of its points. The [[SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[Cartesian product]] of two [[completely orthogonal]] [[great circle]]s. It is a filled [[doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}}
They can be "unrolled" into a square 10×10 array.
Incidentally this structure forms one tetrahedral layer in the [[tetrahedral-octahedral honeycomb]].
There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} How can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}}
In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[triangular bipyramid]] composed of two tetrahedra.

This decomposition of the 600-cell has [[Coxeter notation|symmetry]] [10,2<sup>+</sup>,10], order 400, the same symmetry as the [[grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H<sub>2</sub> × H<sub>2</sub>}}
The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}}

The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells.

===== Boerdijk–Coxeter helix rings =====
The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[Hopf fibration]] which fills the entire 600-cell. Each ring of 30 face-bonded tetrahedra is a cylindrical [[Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension.

{| class="wikitable" width="600"
|[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}}
|[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}}
|[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[Skew polygon|skew]] {30/11} [[star polygon]] with a [[winding number]] of 11 called a [[Triacontagon#Triacontagram|triacontagram<sub>11</sub>]], a continuous tight corkscrew [[helix]] bent into a loop of 30 edges (the magenta edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[Density (polytope)|characterized]] as the [[Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}}
|-
|colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''.
|}

The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three cyan Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}}
The 30 magenta edges joining these vertex pairs form a helical [[Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[Petrie polygon]] is a skew [[Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h<sub>1</sub> h<sub>2</sub>''.}}|name=Petrie polygon in 30-cell ring}}
The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell]], the 600-cell's [[dual polytope]].{{Efn|The [[Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[Petrie polygon]] of the 600-cell and its dual the [[120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}}
The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}}

The 15 orange edges and 15 yellow edges form separate 15-gram helices.
Each orange or yellow edge crosses between two cyan great decagons.
Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}}
Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}}
The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between.
A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop.
The two connected loops forms a single [[Möbius loop]], a skew {15/2} [[pentadecagram]].
The pentadecagram is not shown in these illustrations (but [[#Decagons and 5𝝅 polygrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations.
Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}}
Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}}
They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}}

Five of these 30-cell [[Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H<sub>2</sub> × H<sub>2</sub>}}
Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[chiral]] objects; they either spiral clockwise (right) or counterclockwise (left).
The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings.
Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]].
Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration.
In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings.
Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}}

==== Radial golden triangles ====
The 600-cell can be constructed radially from 720 golden triangles of edge lengths {{radic|0.𝚫}} {{radic|1}} {{radic|1}} which meet at the center of the 4-polytope, each contributing two {{radic|1}} radii and a {{radic|0.𝚫}} edge.
They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral {{radic|0.𝚫}} bases (the faces of the 600-cell).
These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular {{radic|0.𝚫}} tetrahedron bases (the cells of the 600-cell).

==== Characteristic orthoscheme ====
{| class="wikitable floatright"
!colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
|-
!align=right|𝒍
|align=center|<small><math>\tfrac{1}{\phi} \approx 0.618</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
|align=center|<small>164°29′</small>
|align=center|<small><math>\pi-2\text{𝟁}</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>\sqrt{\tfrac{1}{2\phi^2}} \approx 0.437</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>36°</small>
|align=center|<small><math>\tfrac{\pi}{5}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{4\phi^2}} \approx 0.535</math></small>
|align=center|<small>22°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{3} - \text{𝜼}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_1R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{4\phi^2}} \approx 0.309</math></small>
|align=center|<small>18°</small>
|align=center|<small><math>\tfrac{\pi}{10}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_2R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{12\phi^2}} \approx 0.178</math></small>
|align=center|<small>17°44′40″</small>
|align=center|<small><math>\text{𝜼} - \tfrac{\pi}{6}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R^4/l</math></small>
|align=center|<small><math>1</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}} \approx 0.951</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_3R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{\phi^4}{8}} \approx 0.926</math></small>
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!align=right|<small><math>\text{𝜼}</math></small>
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|align=center|<small>37°44′40″</small>
|align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small>
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Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|An [[orthoscheme]] is a [[chiral]] irregular [[simplex]] with [[right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[facet (geometry)|facet]]s (its ''mirror walls'').
Every regular polytope can be dissected radially into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center.
The characteristic orthoscheme has the shape described by the same [[Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}
The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[Coxeter-Dynkin diagram]] {{CDD|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets.
It is an irregular [[Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]].
The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius.
The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center.
Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme.
The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 600-cell has unit radius and edge length <small><math>\text{𝒍} = \tfrac{1}{\phi} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{5 + \sqrt{5}}{8}}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell).
The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.

==== Reflections ====
The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[polychoron]] consists of 3-dimensional cells.}}
Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections.
Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}}
For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram<sub>2</sub> geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}}
Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q<sup>''q''</sup> R<sup>''r''</sup> T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q<sup>2</sup> is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q<sup>''q''</sup> R<sup>''r''</sup><br>
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q<sup>''q''</sup> R<sup>''r''</sup> T<br>
where 2''q'' + ''r'' + 1 ≤ ''n''.<br>
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q<sup>2</sup>, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}}

==== Weyl orbits ====
Another construction method uses [[#Symmetries|quaternions]] and the [[Icosahedral symmetry]] of [[Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4):
: O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
: O(1000) : V1
: O(0010) : V2
: O(0001) : V3

[[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]]

With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell]] of order 14400.

Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct:

* the [[snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math>
* the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> 
* the [[120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math>

=== Rotations ===
The [[#Geometry|regular convex 4-polytopes]] are an [[Group action|expression]] of their underlying [[Symmetry (geometry)|symmetry]] which is known as [[SO(4)]], the [[Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨<sub>4</sub>.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{Efn|A [[Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one [[completely orthogonal]] invariant plane rotates.
Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions).
Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles.
A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''.
Simple rotations are not commutative; left and right rotations (in general) reach different destinations.
The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles.
The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}}
Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by [[Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}}
An '''isoclinic rotation''' is a different special case, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[8-cell#Radial equilateral symmetry|4-dimensional diagonal]].{{Efn|name=isoclinic geodesic}}
The point is displaced a total [[Pythagorean distance]] equal to the square root of four times the square of that distance.
All vertices are displaced to a vertex at least two edge-lengths away.{{Efn|name=isoclinic rotation to non-adjacent vertices}}
For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex {{radic|1}} (60°) distant, moving {{radic|1/4}} {{=}} 1/2 unit radius in four orthogonal directions.|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.
A '''[[geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points).
Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere).
Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}}
But they are not like 3-dimensional [[screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}}
Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once.
They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}}
They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.
These '''[[#Rotations on polygram isoclines|isoclines]]''' are geodesic 1-dimensional lines embedded in a 4-dimensional space.
On the 3-sphere{{Efn|All isoclines are [[geodesics]], and isoclines on the [[3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in [[chiral]] pairs as [[Villarceau circle]]s on the [[Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}}
A single black or white isocline forms a [[Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}}
The double loop is a true circle in four dimensions.{{Efn|name=one true 5𝝅 circle}}
Even and odd isoclines are also linked, not in a Möbius loop but as a [[Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} the geodesic paths traversed by vertices in an [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]].
They are [[Helix|helices]] bent into a [[Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[Winding number|winding route]] around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}}

The 600-cell is generated by [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{Efn|In a ''[[William Kingdon Clifford|Clifford]] displacement'', also known as an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes{{Efn|name=isoclinic invariant planes}} are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle.
A [[William Kingdon Clifford|Clifford]] displacement is [[8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn||name=isoclinic 4-dimensional diagonal}}
Every plane that is Clifford parallel to one of the completely orthogonal planes is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane rotates sideways.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}}
''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}}

==== Twenty-five 24-cells ====
There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney|Hooker|Johnson|Robinson|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}}
Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.

The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other.
Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart.
In a unit-radius 4-polytope, vertices 120° apart are joined by a {{radic|3}} chord.|name=120° apart}}

The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell|Aravind|2009|loc=§3. The 600-cell|pp=2-5}}
There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells.
The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells.
The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}}}

Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]].
The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane.
There are an infinite number of pairs of [[completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]].
The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}}
In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}}

Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon.
The two parallel decagons and the ten linking edges form a double helix ring.
Three decagons can also be parallel (decagons come in parallel [[Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring.
If the ring is cut and laid out flat in 3-space, it is a [[Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long.
The three Clifford parallel decagons can be seen as the cyan edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]].
Each magenta edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}}
An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}}
Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}}
The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once.
Four paths are right-hand [[Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, and four are left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}|name=rotations to 8 disjoint 24-cells}}
The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.

All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}}
Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared).
For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}}
Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}}
With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Non-disjoint 24-cells are related by a [[Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane.
They are also related by an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving at most 6 other vertices fixed (the vertices that the fixed central plane intersects).
The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great [[digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively.
Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane.
In this simple rotation, the hexagon does not move.
The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}}

There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}}
As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space.
Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}}
''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]].
(There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}}
''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart.
However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart.
The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}}
(There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}}
Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane.
The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}
In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}.
The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}}

On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel.
(24-cells do not have great decagons.)
The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell.
The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]].
Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell.
Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel.

==== Rotations on polygram isoclines ====
The regular convex 4-polytopes each have their characteristic kind of right (and left) [[isoclinic rotation]], corresponding to their characteristic kind of discrete [[Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.'' The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell.

Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[Skew polygon|skew]] ''[[Polygram (geometry)|polygrams]]'' (which trace the paths on the [[Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}}
Since isoclinic rotations are [[chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}}
All the fiber bundles are aspects of the same discrete [[Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations.

Cell rings are another expression of the Hopf fibration.
Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating.
In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}}
The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells.
The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets.

Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers.
A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes.
Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}}

In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia|Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}}
The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not.
The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise.
Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}}
The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[Torsion of a curve|torsion]]).
Pairs of left-handed and right-handed isoclines run through cell rings of this kind.
Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}}
Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively.
Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once.
It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers.

Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle.
There are two ways they can do this: by both rotating in the "same" direction, or by rotating in "opposite" directions (according to the [[right hand rule]] by which we conventionally say which way is "up" on each of the 4 coordinate axes).
The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}
Left and right isoclines are different paths that go to different places.
In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers.

A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation.
Each moving vertex travels along an isocline contained within a (moving) cell ring.
While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}}
The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations.
Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines.
The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}}

A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.)
In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration.

An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices (two or more edge-lengths away){{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex at least two edge-lengths away.
In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is exactly two edge-lengths away along one of several great circle geodesic routes: the opposite vertex of a neighboring cell.
In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron.
|name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind).
A left-right pair of isoclinic rotations constitutes a discrete fibration.
All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}}
The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[geodesic]].

==== Decagons and pentadecagrams ====
The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines.

The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}}
12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[Hopf fibration]].
There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}}
The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons.
The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}}
A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]].
A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration).{{^|Efn|name=Schläfli double six}}
The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether.
Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices.

In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}}
In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon.
In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a [[hexagram]]: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are {{radic|3}} chords of the hexagon instead of {{radic|1}} hexagon edges.{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=polytopes ordered by size and complexity}}
In the {{radic|1}} [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply {{radic|3}} chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic {6/2} hexagram rotation]] both rotate circles of 6 vertices.
The hexagram isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}}
The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of hexagrams]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}}
In the {{radic|2}} [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its {{radic|2}} edges and its {{radic|4}} diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[Winding number|winds]] ''three'' times around the 3-sphere.
The invariant central plane completely orthogonal to each {4} great square is another {4} great square {{radic|4}} distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares.
The 16-cell's [[dual polytope]] the [[8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). 
In the 8-cell this is a rotation of {{radic|1}} × {{radic|3}} great rectangles, and also a rotation of {{radic|4}} axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]].
In the {{radic|0.𝚫}} [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are {{radic|1}} hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}}
The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices.
The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices.
The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but [[120-cell#Chords|its characteristic isoclinic rotation]] takes place in completely orthogonal invariant planes which contain {2} great [[digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell.
The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis.
The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=polytopes ordered by size and complexity}} nested like [[Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} 
This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}}

In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it.
Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}}
The two non-adjacent vertices are linked by a {{radic|1}} chord of the isocline which is a great hexagon edge (a 24-cell edge).
The {{radic|1}} chords of the 30-cell ring (without the {{radic|0.𝚫}} 600-cell edges) form a skew [[triacontagram]]<sub>{30/4}=2{15/2}</sub> which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[pentadecagram]]<sub>2</sub> isoclines.
Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers.
Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors.
Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[chessboard]], '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. (Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.)
Things which have [[chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings found in the [[16-cell#Helical construction|16-cell]] and [[#Boerdijk–Coxeter helix rings|600-cell]].
Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]].
Some things have '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}}
Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}}
The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations.

At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end.
Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}}
Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation.
Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines.
There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell.
Each distinct isocline belongs to 5 fiber bundles.

{| class="wikitable" width="450"
!colspan=4|Three sets of 30-cell ring chords from the same [[orthogonal projection]] viewpoint
|-
![[Pentadecagon#Pentadecagram|Pentadecagram {15/2}]]
![[Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]]
![[Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]]
|-
|colspan=2 align=center|All edges are [[pentadecagram]] isocline chords of length {{radic|1}}, which are also [[24-cell#Hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell.
|colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length {{radic|1.𝚫}} ≈ 1.176. 
|-
|[[File:Regular_star_polygon_15-2.svg|200px]]
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_6(5,1).svg|200px]]
|-
|valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The {{radic|1}} chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart.
|valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The {{radic|1}} chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra.
|valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly congruent, each ''acting'' as a left or right isocline in different fibrations. 
|- 
|colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. 
|}

Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings, but the two isoclines in each 3-cell ring are directly congruent.{{Efn|The chord-path of an isocline may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}
The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit.
The double loop is entirely contained within a single [[#Boerdijk–Coxeter helix rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}}
Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell.
Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[Möbius strip]], exactly one edge length apart.
Thus each cell has two helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points.
Globally these two helices are a single connected circle of ''both'' chiralities,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[Torsion of a curve|torsion]].
An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}}
The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one {{radic|0.𝚫}} edge-length apart).
The 30 chords joining the isocline's 30 vertices are {{radic|1}} hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell {{radic|0.𝚫}} edges apart on a decagon great circle.
{{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram<sub>2</sub> geodesic]] is bent into a twisted ring in the fourth dimension like a [[Möbius strip]], its [[screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right).
The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions.
The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''.
The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon.
The Petrie triacontagon has {{radic|0.𝚫}} edges; the isoclinic pentadecagram<sub>2</sub> has {{radic|1}} edges which join vertices which are two {{radic|0.𝚫}} edges apart.
Each {{radic|1}} edge belongs to a different [[#Hexagons|great hexagon]], and successive {{radic|1}} edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}}
These isocline chords are both hexa''gon'' edges and penta''gram'' edges.

The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other.
Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[chessboard]].{{Efn|name=isoclinic chessboard}}
In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring.

==== Hexagons and hexagrams ====
[[File:Regular_star_figure_2(10,3).svg|thumb|[[Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines.

Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon.
The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black {{radic|3}} [[24-cell#Triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring.
The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 {{radic|1.𝚫}} chords of each isocline form a skew [[Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]].

Notice the relation between the [[24-cell#Helical hexagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on hexagram isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline, and the 600-cell's {{radic|1.𝚫}} isocline chord is shorter than the 24-cell's {{radic|3}} isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on [[24-cell#Helical hexagrams and their 4𝝅 isoclines|hexagrams]], while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference 2𝝅; simple rotations take place on 2𝝅 isoclines.  Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[20-gon|icosagram]] is a compound of the 24-cell's helical {6/2} hexagram, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}}

==== Squares and octagrams ====
[[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[24-gon#Related polygons|{24/5} 24-gram]], with <big>φ</big> {{=}} {{radic|2.𝚽}} edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell ({{radic|1}} edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines.

Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines.

The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <big>φ</big> {{=}} {{radic|2.𝚽}} chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one {{radic|1}} chord apart, and 5 <big>φ</big> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell.

Notice the relations between the [[16-cell#Helical construction|16-cell's rotation of just 2 invariant great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the {{radic|4}} diameter. In the 600-cell vertices are closer together, and its {{radic|2.𝚽}} {{=}} <big>φ</big> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference.

=== As a configuration ===
This [[Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

<math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math>

Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal.
{| class=wikitable
!H<sub>4</sub>||{{CDD|node_1|3|node|3|node|5|node}}
! [[k-face|''k''-face]]||f<sub>''k''</sub>||f<sub>0</sub> || f<sub>1</sub>||f<sub>2</sub>||f<sub>3</sub>||[[vertex figure|''k''-fig]]
!Notes
|- align=right
|H<sub>3</sub> || {{CDD|node_x|2|node|3|node|5|node}} ||( )
!f<sub>0</sub>
|| 120 ||  12 ||   30 ||  20 ||[[icosahedron|{3,5}]] || H<sub>4</sub>/H<sub>3</sub> = 14400/120 = 120
|- align=right
|A<sub>1</sub>H<sub>2</sub> ||{{CDD|node_1|2|node_x|2|node|5|node}} ||{ }
!f<sub>1</sub>
||   2 || 720 ||    5 ||   5 || [[pentagon|{5}]] || H<sub>4</sub>/H<sub>2</sub>A<sub>1</sub> = 14400/10/2 = 720
|- align=right
|A<sub>2</sub>A<sub>1</sub> ||{{CDD|node_1|3|node|2|node_x|2|node}} ||[[equilateral triangle|{3}]]
!f<sub>2</sub>
||   3 ||   3 || 1200 ||   2 || { } || H<sub>4</sub>/A<sub>2</sub>A<sub>1</sub> = 14400/6/2 = 1200
|- align=right
|A<sub>3</sub> ||{{CDD|node_1|3|node|3|node|2|node_x}} ||[[tetrahedron|{3,3}]]
!f<sub>3</sub>
||   4 ||   6 ||    4 || 600|| ( ) || H<sub>4</sub>/A<sub>3</sub> = 14400/24 = 600
|}

== Symmetries ==
The [[icosian]]s are a specific set of Hamiltonian [[quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}}
The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[rational number]]s.{{Sfn|Steinbach|1997|p=24}}
The finite sums of the 120 [[Icosian#Unit icosians|unit icosians]] are called the [[Icosian#Icosian ring|icosian ring]].

When interpreted as [[quaternion]]s,{{Efn|In [[Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[quaternion]] is simply a (w, x, y, z) Cartesian coordinate.|name=quaternions}} the 120 vertices of the 600-cell form a [[group (mathematics)|group]] under quaternionic multiplication.
This group is often called the [[binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}}
It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[invariant subgroup]], namely as the subgroup ''2I<sub>L</sub>'' of quaternion left-multiplications and as the subgroup ''2I<sub>R</sub>'' of quaternion right-multiplications.
Each rotational symmetry of the 600-cell is generated by specific elements of ''2I<sub>L</sub>'' and ''2I<sub>R</sub>''; the pair of opposite elements generate the same element of ''RSG''.
The [[Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''.
We have the isomorphism ''RSG ≅ (2I<sub>L</sub> × 2I<sub>R</sub>) / {Id, -Id}''.
The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200.
The [[quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|2015|p=1|loc="''[[Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}}

The binary icosahedral group is [[isomorphic]] to [[special linear group|SL(2,5)]].

The full [[symmetry group]] of the 600-cell is the [[H4 (mathematics)|Coxeter group H<sub>4</sub>]].{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§2 The Labeling of H<sub>4</sub>}}
This is a [[group (mathematics)|group]] of order 14400.
It consists of 7200 [[Rotation (mathematics)|rotations]] and 7200 rotation-reflections.
The rotations form an [[invariant subgroup]] of the full symmetry group.
The rotational symmetry group was first described by S.L. van Oss.{{Sfn|Oss|1899||pp=1-18}}
The H<sub>4</sub> group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'.
In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly.
Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes....
This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}}

== Visualization ==
The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}}  One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections.

=== 2D projections ===
The H3 [[decagon]]al projection shows the plane of the [[van Oss polygon]].

{| class="wikitable" width=600
|+ [[Orthographic projection]]s by [[Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
|- align=center
!H<sub>4</sub>
! -
!F<sub>4</sub>
|- align=center
|[[File:600-cell graph H4.svg|200px]]<br>[30]<br>(Red=1)
|[[File:600-cell t0 p20.svg|200px]]<br>[20]<br>(Red=1)
|[[File:600-cell t0 F4.svg|200px]]<br>[12]<br>(Red=1)
|- align=center
!H<sub>3</sub>
!A<sub>2</sub> / B<sub>3</sub> / D<sub>4</sub>
!A<sub>3</sub> / B<sub>2</sub>
|- align=center
|[[File:600-cell t0 H3.svg|200px]]<br>[10]<br>(Red=1,orange=5,yellow=10)
|[[File:600-cell t0 A2.svg|200px]]<br>[6]<br>(Red=1,orange=3,yellow=6)
|[[File:600-cell t0.svg|200px]]<br>[4]<br>(Red=1,orange=2,yellow=4)
|}

=== 3D projections ===
A three-dimensional model of the 600-cell, in the collection of the [[Institut Henri Poincaré]], was photographed in 1934–1935 by [[Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.<ref>{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&nbsp;58; ''Antony and Cleopatra'', SE-6, p.&nbsp;59; ''mathematical object mo-9'', p.&nbsp;64; ''Merchant of Venice'', SE-9, p.&nbsp;65, and "The Hexacosichoron", Philip Ordning, p.&nbsp;96.</ref>

{| class=wikitable
!colspan=2|Vertex-first projection
|-
|[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]]
|This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
* The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
* The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
* The remaining cells are rendered in edge-outline.
* Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image.
|-
!colspan=2|Cell-first projection
|-
|[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]]
|This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
* The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
* The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
* The remaining cells are rendered in edge-outline.
* Cells facing away from the 4D viewpoint have been culled for clarity.

This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
|}

{| class=wikitable
!Frame synchronized orthogonal isometric (left) and perspective (right) projections
|-
|[[File:Cell600Cmp.ogv|640px]]
|}

== Diminished 600-cells ==
The [[snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell]] and taking the [[convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[Diminishment (geometry)|diminishing]]'' of the 600-cell.

The [[grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H<sub>2</sub> × H<sub>2</sub>}}

A bi-24-diminished 600-cell, with all [[tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.

There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.<ref>{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}</ref>
{| class="wikitable collapsible"
!colspan=12|Diminished 600-cells
|-
!Name
!Tri-24-diminished 600-cell
!Bi-24-diminished 600-cell
![[Snub 24-cell]]<br>(24-diminished 600-cell)
![[Grand antiprism]]<br>(20-diminished 600-cell)
!600-cell
|- align=center
!Vertices
|48
|72
|96
|100
|120
|- align=center
!Vertex figure<br>(Symmetry)
|[[File:Dual tridiminished icosahedron.png|120px]]<br>dual of tridiminished icosahedron<br>([3], order 6)
|[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]<br>[[Hexahedron|tetragonal antiwedge]]<br>([2]<sup>+</sup>, order 2)
|[[File:Snub 24-cell verf.png|120px]]<br>[[tridiminished icosahedron]]<br>([3], order 6)
|[[File:Grand antiprism verf.png|120px]]<br>[[Edge-contracted icosahedron|bidiminished icosahedron]]<br>([2], order 4)
|[[File:600-cell verf.svg|120px]]<br>[[Icosahedron]]<br>([5,3], order 120)
|- align=center
!Symmetry
|colspan=2|Order 144 (48×3 or 72×2)
|[3<sup>+</sup>,4,3]<br>Order 576 (96×6)
|[10,2<sup>+</sup>,10]<br>Order 400 (100×4)
|[5,3,3]<br>Order 14400 (120×120)
|- align=center
!Net
|[[File:Triicositetradiminished hexacosichoron net.png|100px]]
|[[File:Biicositetradiminished hexacosichoron net.png|100px]]
|[[File:Snub 24-cell-net.png|100px]]
|[[File:Grand antiprism net.png|100px]]
|[[File:600-cell net.png|100px]]
|- align=center
!Ortho<br>H<sub>4</sub> plane
|[[File:Tridiminished 600-cell H4 Coxeter plane.svg|120px]]
|[[File:bidex ortho-30-gon.png|120px]]
|[[File:Snub 24-cell ortho30-gon.png|120px]]
|[[File:Grand antiprism ortho-30-gon.png|120px]]
|[[File:600-cell graph H4.svg|120px]]
|- align=center
!Ortho<br>F<sub>4</sub> plane
|[[File:Tridiminished 600-cell F4 Coxeter plane.svg|120px]]
|[[File:Bidex ortho 12-gon.png|120px]]
|[[File:24-cell h01 F4.svg|120px]]
|[[File:GrandAntiPrism-2D-F4.svg|120px]]
|[[File:600-cell t0 F4.svg|120px]]
|}

== Related polytopes and honeycombs ==
The 600-cell is one of 15 regular and uniform polytopes with the same H<sub>4</sub> symmetry [3,3,5]:{{Sfn|Denney|Hooker|Johnson|Robinson|2020}}
{{H4_family}}

It is similar to three [[regular 4-polytope]]s: the [[5-cell]] {3,3,3}, [[16-cell]] {3,3,4} of Euclidean 4-space, and the [[order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[tetrahedron|tetrahedral]] cells.
{{Tetrahedral cell tessellations}}

This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[icosahedron]] vertex figures:
{{Icosahedral vertex figure tessellations}}

The [[regular complex polytope|regular complex polygons]] <sub>3</sub>{5}<sub>3</sub>, {{CDD|3node_1|5|3node}} and <sub>5</sub>{3}<sub>5</sub>, {{CDD|5node_1|3|5node}}, in <math>\mathbb{C}^2</math> have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[Complex reflection group]] <sub>3</sub>[5]<sub>3</sub>, order 360, and the second has symmetry <sub>5</sub>[3]<sub>5</sub>, order 600.{{Sfn|Coxeter|1991|pp=48-49}}

{| class="wikitable mw-collapsible mw-collapsed"
!colspan=3| Regular complex polytope in orthogonal projection of H<sub>4</sub> Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
|- align=center
|[[File:600-cell graph H4.svg|240px]]<br>{3,3,5}<br>Order 14400
|[[File:Complex polygon 3-5-3.png|240px]]<br><sub>3</sub>{5}<sub>3</sub><br>Order 360
|[[File:Complex polygon 5-3-5.png|240px]]<br><sub>5</sub>{3}<sub>5</sub><br>Order 600
|}

== See also ==
* [[24-cell]], the predecessor [[4-polytope]] on which the 600-cell is based
* [[120-cell]], the dual [[4-polytope]] to the 600-cell, and its successor
* [[Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]]
* [[Regular 4-polytope]]
* [[Polytope]]

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}

== External links ==
* {{MathWorld | urlname = 600-Cell | title = 600-Cell}}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o5o - ex}}
* [http://www.polytope.de/c600.html Der 600-Zeller (600-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
* [http://eusebeia.dyndns.org/4d/600-cell The 600-Cell] Vertex centered expansion of the 600-cell

{{Regular 4-polytopes}}
{{Polytopes}}
{{Authority control}}

[[Category:Regular 4-polytopes|600]]
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