View source for 5-cell

{{Short description|Four-dimensional analogue of the tetrahedron}}
{{For|the sequence of fifth element numbers of Pascal's triangle|Pentatope number}}
{{more sources needed|date=August 2024}}
{{excessive detail|date=August 2024}}
{{Infobox polychoron
 | Name=5-cell<BR>(4-simplex)
 | Image_File=5-cell.gif
 | Image_Caption=A 3D orthogonal projection of a 5-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]]
 | Type=[[Convex regular 4-polytope]]
 | Family=[[Simplex]]
 | Last= 
 | Index=1
 | Next=[[Rectified 5-cell|2]]
 | Schläfli={3,3,3}
 | CD={{CDD|node_1|3|node|3|node|3|node}}
 | Cell_List=5 [[tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] 
 | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]
 | Edge_Count= 10
 | Vertex_Count= 5
 | Petrie_Polygon=[[pentagon]]
 | Coxeter_Group= A<sub>4</sub>, [3,3,3]
 | Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[tetrahedron]])
 | Dual=[[Self-dual polytope|Self-dual]]
 | Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
}}

In [[geometry]], the '''5-cell''' is the convex [[4-polytope]] with [[Schläfli symbol]] {3,3,3}. It is a 5-vertex [[four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells. It is also known as a '''C<sub>5</sub>''', '''hypertetrahedron''', '''pentachoron''',{{sfn|Johnson|2018|p=249}} '''pentatope''', '''pentahedroid''',{{sfn|Ghyka|1977|p=[https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA68 68]}} '''tetrahedral pyramid''', or '''4-[[simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[tetrahedron]] in three dimensions and the [[triangle]] in two dimensions. The 5-cell is a [[Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides.

The '''regular 5-cell''' is bounded by five [[regular tetrahedron|regular tetrahedra]], and is one of the six [[regular convex 4-polytope]]s (the four-dimensional analogues of the [[Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions.

== Properties ==
The 5-cell is the 4-dimensional [[simplex]], the simplest possible [[4-polytope]]. In other words, the 5-cell is a [[polychoron]] analogous to a [[tetrahedron]] in high dimension.{{sfn|Miyazaki|Ishii|2021|p=[https://books.google.com/books?id=d2VXEAAAQBAJ&pg=PA46 46]}} It is formed by any five points which are not all in the same [[hyperplane]] (as a tetrahedron is formed by any four points which are not all in the same plane, and a [[triangle]] is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The ''regular'' 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex [[120-cell]] is a [[Polytope compound|compound]] of 120 regular 5-cells.

The 5-cell is [[Self-dual polytope|self-dual]], meaning its dual polytope is 5-cell itself.{{sfn|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA41 41]}} Its maximal intersection with 3-dimensional space is the [[triangular prism]]. Its dichoral angle is <math display=inline> \arccos(-1/4) \approx 75.52^\circ </math>.{{sfn|Akiyama|Hitotumatu|Sato|2012}}

It is the first in the sequence of 6 convex regular 4-polytopes, in order of volume at a given radius or number of vertexes.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions}}

The convex hull of two 5-cells in dual configuration is the [[Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], dual of the [[Truncated 5-cell#Bitruncated 5-cell|bitruncated 5-cell]].

=== As a configuration ===
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 5-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 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.{{sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} The ''k''-faces can be read as rows left of the diagonal, while the ''k''-figures are read as rows after the diagonal.<ref>{{cite web | url=https://bendwavy.org/klitzing/incmats/pen.htm | title=Pen }}</ref>
[[File:Symmetrical 5-set Venn diagram.svg|thumb|Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975]]
{| class=wikitable
|-
!Element||''k''-face||f<sub>''k''</sub>
!f<sub>0</sub>
!f<sub>1</sub>
!f<sub>2</sub>
!f<sub>3</sub>
!''k''-figs
|- align=right
|align=left bgcolor=#ffffe0 |<!--. . . .-->{{CDD|node_x|2|node_x|2|node_x|2|node_x}}||( )
|rowspan=1|f<sub>0</sub>
|bgcolor=#e0ffe0|5
|  bgcolor=#e0e0e0|4
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|4
|[[Tetrahedron|{3,3}]]
|- align=right
|align=left bgcolor=#ffffe0 |<!--x . . .-->{{CDD|node_1|2|node_x|2|node_x|2|node_x}}||{ }
|rowspan=1|f<sub>1</sub>
|  bgcolor=#e0e0e0|2
|bgcolor=#e0ffe0|10
|  bgcolor=#e0e0e0|3
|  bgcolor=#ffffff|3
|[[Triangle|{3}]]
|- align=right
|align=left bgcolor=#ffffe0 |<!--x3o . .-->{{CDD|node_1|3|node|2|node_x|2|node_x}}||[[triangle|{3}]]
|rowspan=1|f<sub>2</sub>
|  bgcolor=#ffffff|3
|  bgcolor=#e0e0e0|3
|bgcolor=#e0ffe0|10
|  bgcolor=#e0e0e0|2
|{ }
|- align=right
|align=left bgcolor=#ffffe0 |<!--x3o3o .-->{{CDD|node_1|3|node|3|node|2|node_x}}||[[Tetrahedron|{3,3}]]
|rowspan=1|f<sub>3</sub>
|  bgcolor=#e0e0e0|4
|  bgcolor=#ffffff|6
|  bgcolor=#e0e0e0|4
|bgcolor=#e0ffe0|5
|( )
|}
All these elements of the 5-cell are enumerated in [[Branko Grünbaum]]'s [[Venn diagram]] of 5 points, which is literally an illustration of the regular 5-cell in [[#Projections|projection]] to the plane.



=== Geodesics and rotations ===
[[File:5-cell-orig.gif|thumb|A 3D projection of a 5-cell performing a [[SO(4)#Double rotations|double rotation]].]]
The 5-cell has only [[digon]] central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]] of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no vertices of the 5-cell.

Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The [[Clifford torus]] is depicted in its rectangular (wrapping) form.

<gallery caption="[[Rotations in 4-dimensional Euclidean space#Visualization of 4D rotations|Visualization of 4D rotations]]">
File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane
File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane
File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio
File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation
File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation
</gallery>

===Projections===
[[Image:Stereographic polytope 5cell.png|240px|thumb|[[Stereographic projection]] wireframe (edge projected onto a [[3-sphere]])]]

The A<sub>4</sub> Coxeter plane projects the 5-cell into a regular [[pentagon]] and [[pentagram]]. The A<sub>3</sub> Coxeter plane projection of the 5-cell is that of a [[square pyramid]]. The A<sub>2</sub> Coxeter plane projection of the regular 5-cell is that of a [[triangular bipyramid]] (two tetrahedra joined face-to-face) with the two opposite vertices centered.

{{4-simplex Coxeter plane graphs|t0|150}}

{|class="wikitable" width=640
!colspan=2|Projections to 3 dimensions
|- valign=top align=center
|[[Image:Pentatope-vertex-first-small.png]]<BR>The vertex-first projection of the 5-cell into 3 dimensions has a [[tetrahedron|tetrahedral]] projection envelope. The closest vertex of the 5-cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.
|[[Image:5cell-edge-first-small.png]]<BR>The edge-first projection of the 5-cell into 3 dimensions has a [[triangular dipyramid]]al envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope.
|- valign=top align=center
|[[Image:5cell-face-first-small.png]]<BR>The face-first projection of the 5-cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.
|[[Image:5cell-cell-first-small.png|320px]]<BR>The cell-first projection of the 5-cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.
|}

== Irregular 5-cells ==
In the case of [[Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[fundamental domain]]s of the different [[Coxeter group|symmetry groups]] which give rise to the various 4-polytopes.

===Orthoschemes===
A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[Triangle#By_internal_angles|right triangles]]. (The 5 vertices form 5 tetrahedral [[Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.) An [[Schläfli orthoscheme|orthoscheme]] is an irregular [[simplex]] that is the [[convex hull]] of a [[Tree (graph theory)|tree]] in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle).

Orthoschemes are the [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]].

[[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes). The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge.

The 4-cube {{CDD|node_1|4|node|3|node|3|node}} can be [[Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{CDD|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.

More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets. More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[genetic code]]s of polytopes: like a [[Swiss Army knife]], they contain one of everything needed to construct the polytope by replication.

Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme. There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[Hyperpyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{CDD|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{CDD|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼<sub>4</sub>"}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
|-
!align=right|𝒍
|align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small>
|align=center|<small>104°30′40″</small>
|align=center|<small><math>\pi - 2\text{𝜂}</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>\pi - 2\text{𝟁}</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>2\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}{30}} \approx 0.183</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</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}{20}} \approx 0.387</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>2\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}{20}} \approx 0.224</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</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}{60}} \approx 0.129</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</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>\sqrt{1} = 1.0</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{3}{8}} \approx 0.612</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{1}{6}} \approx 0.408</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{1}{16}} = 0.25</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
|
|
|
|
|-
!align=right|<small><math>\text{𝜼}</math></small>
|align=center|
|align=center|<small>37°44′40″</small>
|align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small>
|align=center|
|align=center|
|}
The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell). 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. If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</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}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</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>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.

===Isometries===
There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[vertex figure]]s:
{| class=wikitable width=600
!Symmetry
![3,3,3]<BR>Order 120
![3,3,1]<BR>Order 24
![3,2,1]<BR>Order 12
![3,1,1]<BR>Order 6
!~[5,2]<sup>+</sup><BR>Order 10
|- align=center
!Name
| Regular 5-cell
| Tetrahedral [[polyhedral pyramid|pyramid]]
| 
| Triangular pyramidal pyramid
|
|- align=center
![[Schläfli symbol|Schläfli]]
| {3,3,3}
| {3,3}∨(&nbsp;)
| {3}∨{&nbsp;}
| {3}∨(&nbsp;)∨(&nbsp;)
| 
|- align=center valign=top
!valign=center|Example<BR>Vertex<BR>figure
|[[File:5-simplex verf.png|120px]]<BR>[[5-simplex]]
|[[File:Truncated 5-simplex verf.png|120px]]<BR>[[Truncated 5-simplex]]
|[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[Bitruncated 5-simplex]]
|[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[Cantitruncated 5-simplex]]
|[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[Omnitruncated 4-simplex honeycomb]]
|}

The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[polyhedral pyramid]], constructed as a regular [[tetrahedron]] base in a 3-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[triangular pyramid]] cells.

Many [[uniform 5-polytope]]s have '''tetrahedral pyramid''' [[vertex figure]]s with [[Schläfli symbol]]s ( )∨{3,3}.
{| class=wikitable
|+ Symmetry [3,3,1], order 24
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:5-cell prism verf.png|100px]]
|[[File:Tesseractic prism verf.png|100px]]
|[[File:120-cell prism verf.png|100px]]
|[[File:Truncated 5-simplex verf.png|100px]]
|[[File:Truncated 5-cube verf.png|100px]]
|[[File:Truncated 24-cell honeycomb verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[5-cell prism|{ }×{3,3,3}]]<BR>{{CDD|node_1|2|node_1|3|node|3|node|3|node}}
![[Tesseractic prism|{ }×{4,3,3}]]<BR>{{CDD|node_1|2|node_1|4|node|3|node|3|node}}
![[120-cell prism|{ }×{5,3,3}]]<BR>{{CDD|node_1|2|node_1|5|node|3|node|3|node}}
![[Truncated 5-simplex|t{3,3,3,3}]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}}
![[Truncated 5-cube|t{4,3,3,3}]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}}
![[Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{CDD|node_1|3|node_1|4|node|3|node|3|node}}
|}

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram.
{| class=wikitable
!Symmetry
!colspan=2|[3,2,1], order 12
!colspan=2|[3,1,1], order 6
![2<sup>+</sup>,4,1], order 8
![2,1,1], order 4
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Bitruncated 5-simplex verf.png|100px]]
|[[File:Bitruncated penteract verf.png|100px]]
|[[File:Canitruncated 5-simplex verf.png|100px]]
|[[File:Canitruncated 5-cube verf.png|100px]]
|[[File:Bicanitruncated 5-simplex verf.png|100px]]
|[[File:Bicanitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[bitruncated 5-simplex|t<sub>12</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node}}
![[bitruncated 5-cube|t<sub>12</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-simplex|t<sub>012</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-cube|t<sub>012</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node}}
![[Bicantitruncated 5-simplex|t<sub>123</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}
![[Bicantitruncated 5-cube|t<sub>123</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node_1|3|node}}
|}

{| class=wikitable
!Symmetry
!colspan=3|[2,1,1], order 2
![2<sup>+</sup>,1,1], order 2
![ ]<sup>+</sup>, order 1
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Runcicantitruncated 5-simplex verf.png|100px]]
|[[File:Runcicantitruncated 5-cube verf.png|100px]]
|[[File:Runcicantitruncated 5-orthoplex verf.png|100px]]
|[[File:Omnitruncated 5-simplex verf.png|100px]]
|[[File:Omnitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[Runcicantitruncated 5-simplex|t<sub>0123</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-cube|t<sub>0123</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-orthoplex|t<sub>0123</sub>β<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|4|node}}
![[Omnitruncated 5-simplex|t<sub>01234</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
![[Omnitruncated 5-cube|t<sub>01234</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}
|}

== Construction ==
=== As a Boerdijk–Coxeter helix ===
[[File:5-cell 5-ring net.png|480px|thumb|The 5-cell Boerdijk&ndash;Coxeter helix]]
A 5-cell can be constructed as a [[Boerdijk–Coxeter helix]] of five chained tetrahedra, folded into a 4-dimensional ring.{{Sfn|Banchoff|2013}}{{Failed verification|date=March 2025|reason=Banchoff 2013 describes the decomposition of the 8-cell and 24-cell into tori, but neither discusses the 5-cell nor anything relating to Boerdijk–Coxeter helices.}} The 10 triangle faces can be seen in a 2D net within a [[triangular tiling]], with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges form a [[pentagon#Regular pentagons|regular pentagon]] which is the [[Petrie polygon]] of the 5-cell. The blue edges connect every second vertex, forming a [[pentagram]] which is the ''Clifford polygon'' of the 5-cell. The pentagram's blue edges are chords of the 5-cell's ''isocline'', the circular rotational path its vertices take during an [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]], also known as a [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]].

=== Net ===
[[File:5-cell net.png|thumb|right|[[Net (polyhedron)|Net of five tetrahedra (one hidden)]]]]
When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges, and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. This makes the six tetrahedron as its [[Cell (geometry)|cell]].{{sfn|Akiyama|Hitotumatu|Sato|2012}}

===Coordinates===
The simplest set of [[Cartesian coordinates]] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (𝜙,𝜙,𝜙,𝜙), with edge length 2{{sqrt|2}}, where 𝜙 is the [[golden ratio]].{{sfn|Coxeter|1991|p=30|loc=§4.2. The Crystallographic regular polytopes}} While these coordinates are not origin-centered, subtracting <math>(1,1,1,1)/(2-\tfrac{1}{\phi})</math> from each translates the 4-polytope's [[circumcenter]] to the origin with radius <math>2(\phi-1/(2-\tfrac{1}{\phi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888</math>, with the following coordinates:

:<math>\left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,\tfrac{2}{\phi}-3,1,1 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,1,\tfrac{2}{\phi}-3,1 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(1,1,1,\tfrac{2}{\phi}-3 \right)/(\tfrac{1}{\phi}-2)</math>
:<math>\left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi}  \right)/(\tfrac{1}{\phi}-2)</math>

The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a [[Tetrahedron#Coordinates for a regular tetrahedron|regular tetrahedral base]] in 3-space:

:<math>\left( 1, 1, 1, \frac{-1}\sqrt{5}\right)</math>
:<math>\left( 1,-1,-1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left(-1, 1,-1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left(-1,-1, 1,\frac{-1}\sqrt{5} \right)</math>
:<math>\left( 0, 0, 0,\frac{4}\sqrt{5}  \right)</math>

Scaling these or the previous set of coordinates by <math>\tfrac{\sqrt{5}}{4}</math> give '''''unit-radius''''' origin-centered regular 5-cells with edge lengths <math>\sqrt{\tfrac{5}{2}}</math>. The hyperpyramid has coordinates:

:<math>\left( \sqrt{5}, \sqrt{5}, \sqrt{5},  -1 \right)/4</math>
:<math>\left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4</math>
:<math>\left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4</math>
:<math>\left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4</math>
:<math>\left( 0, 0, 0, 1 \right)</math>

Coordinates for the vertices of another origin-centered regular 5-cell with edge length&nbsp;2 and radius <math>\sqrt{\tfrac{8}{5}}\approx 1.265</math> are:

:<math>\left( \frac{1}{\sqrt{10}},\  \frac{1}{\sqrt{6}},\  \frac{1}{\sqrt{3}},\  \pm1\right)</math>
:<math>\left( \frac{1}{\sqrt{10}},\  \frac{1}{\sqrt{6}},\  \frac{-2}{\sqrt{3}},\ 0   \right)</math>
:<math>\left( \frac{1}{\sqrt{10}},\  -\sqrt{\frac{3}{2}},\ 0,\                   0   \right)</math>
:<math>\left( -2\sqrt{\frac{2}{5}},\ 0,\                   0,\                   0   \right)</math>

Scaling these by <math>\sqrt{\tfrac{5}{8}}</math> to unit-radius and edge length <math>\sqrt{\tfrac{5}{2}}</math> gives:
:<math>\left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3})</math>
:<math>\left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3})</math>
:<math>\left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})</math>
:<math>\left(-1, 0, 0, 0 \right)</math>

The vertices of a 4-simplex (with edge {{radic|2}} and radius 1) can be more simply constructed on a [[hyperplane]] in 5-space, as (distinct) permutations of (0,0,0,0,1) ''or'' (0,1,1,1,1); in these positions it is a [[facet (geometry)|facet]] of, respectively, the [[5-orthoplex]] or the [[rectified penteract]].


== Compound ==
The compound of two 5-cells in dual configurations can be seen in this A5 [[Coxeter plane]] projection, with a red and blue 5-cell vertices and edges. This compound has <nowiki>[[3,3,3]]</nowiki> symmetry, order 240. The intersection of these two 5-cells is a uniform [[bitruncated 5-cell]]. {{CDD|branch_11|3ab|nodes}} = {{CDD|branch|3ab|nodes_10l}} ∩ {{CDD|branch|3ab|nodes_01l}}.
:[[File:Compound_dual_5-cells_A5_coxeter_plane.png|240px]]

This compound can be seen as the 4D analogue of the 2D [[hexagram]] &lbrace;{{sfrac|6|2}}&rbrace; and the 3D [[compound of two tetrahedra]].

== Related polytopes and honeycombs ==
The pentachoron (5-cell) is the simplest of 9 [[Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[Coxeter group]].
{{Pentachoron family small}}

{{1 k2 polytopes}}

{{2 k1 polytopes}}

It is in the {p,3,3} sequence of [[regular polychora]] with a [[tetrahedron|tetrahedral]] [[vertex figure]]: the [[tesseract]] {4,3,3} and [[120-cell]] {5,3,3} of Euclidean 4-space, and the [[hexagonal tiling honeycomb]] {6,3,3} of hyperbolic space.
{{Tetrahedral vertex figure tessellations small}}

It is one of three {3,3,p} [[regular 4-polytope]]s with tetrahedral cells, along with the [[16-cell]] {3,3,4} and [[600-cell]] {3,3,5}. The [[order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space also has tetrahedral cells.
{{Tetrahedral cell tessellations}}

It is self-dual like the [[24-cell]] {3,4,3}, having a [[palindromic]] {3,p,3} [[Schläfli symbol]].
{{Symmetric_tessellations}}

{{Symmetric2_tessellations}}

== Notes ==
{{Notelist|40em}}

== Citations ==
{{Reflist}}

== References ==
* {{cite journal
 | last1 = Akiyama | first1 = Jin
 | last2 = Hitotumatu | first2 = Sin
 | last3 = Sato | first3 = Ikuro
 | title = Determination of the element numbers of the regular polytopes
 | journal = Geometriae Dedicata
 | doi = 10.1007/s10711-011-9647-3
 | volume = 159 | pages = 89–97 | year = 2012
}}
* {{cite book
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ
}}
* {{cite book
 | last = Ghyka | first = Matila
 | title = The Geometry of Art and Life
 | year = 1977
 | publisher = [[Dover Publications]]
 | isbn = 978-0-486-23542-4
 | url = https://books.google.com/books?id=h1Wm6tHAKHcC
}}
* {{cite book
 | last = Johnson | first = N. W. | author-link = Norman Johnson (mathematician)
 | title = Geometries and Transformations
 | year = 2018
 | isbn = 978-1-107-10340-5
 | chapter = Finite Symmetry Groups, 11.5 Spherical Coxeter groups
| publisher = Cambridge University Press }}
* {{cite book
 | last1 = Miyazaki | first1 = K.
 | last2 = Ishii | first2 = M.
 | editor-last = Darvas | editor-first = György
 | title = Complex Symmetries
 | url = https://books.google.com/books?id=d2VXEAAAQBAJ
 | contribution = Symmetry in Projection of 4-Dimensional Regular Polychora
 | doi = 10.1007/978-3-030-88059-0
 | year = 2021
 | isbn = 978-3-030-88059-0
}}
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
*** p.&nbsp;120, §7.2. see illustration Fig 7.2<small>A</small>
*** p.&nbsp;296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** {{Citation | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }}
** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}}
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;409: Hemicubes: 1<sub>n1</sub>)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}}
==External links==
* {{MathWorld|urlname=Pentatope|title=Pentatope}}
* {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o3o - pen}}
* [http://www.polytope.de/c5.html Der 5-Zeller (5-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
* [http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, Regular polychora]
* [https://web.archive.org/web/20110718202453/http://public.beuth-hochschule.de/~meiko/pentatope.html Java3D Applets]
* [http://hi.gher.space/wiki/Pyrochoron pyrochoron]

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

[[Category:Regular 4-polytopes]]
[[Category:Pyramids (geometry)]]