Regular 4-polytope

File:Hypercube.svg

The tesseract is one of 6 convex regular 4-polytopes

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

There are six convex and ten star regular 4-polytopes, giving a total of sixteen.

History

The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.Template:Sfn He discovered that there are precisely six such figures.

Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2). That excludes cells and vertex figures such as the great dodecahedron {5,Template:Sfrac} and small stellated dodecahedron {Template:Sfrac,5}.

Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.

Construction

The existence of a regular 4-polytope $\{p,q,r\}$ is constrained by the existence of the regular polyhedra $\{p,q\},\{q,r\}$ which form its cells and a dihedral angle constraint

$\sin \frac{\pi }{p}\sin \frac{\pi }{r}>\cos \frac{\pi }{q}$

to ensure that the cells meet to form a closed 3-surface.

The six convex and ten star polytopes described are the only solutions to these constraints.

There are four nonconvex Schläfli symbols {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,Template:Sfrac,3}, {4,3,Template:Sfrac}, {Template:Sfrac,3,4}, {Template:Sfrac,3,Template:Sfrac}.

Regular convex 4-polytopes

The regular convex 4-polytopes are the four-dimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.

Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).

Properties

Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content within the same radius.Template:Sfn The 4-simplex (5-cell) has the smallest content, and the 120-cell has the largest.

The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.

Names	Image	Family	Schläfli Coxeter	V	E	F	C	Vert. fig.	Dual	Symmetry group
5-cell pentachoron pentatope 4-simplex	File:4-simplex t0.svg	n-simplex (An family)	{3,3,3} Template:CDD	5	10	10 {3}	5 {3,3}	{3,3}	self-dual	A4 [3,3,3]	120
16-cell hexadecachoron 4-orthoplex	File:4-cube t3.svg	n-orthoplex (Bn family)	{3,3,4} Template:CDD	8	24	32 {3}	16 {3,3}	{3,4}	8-cell	B4 [4,3,3]	384
8-cell octachoron tesseract 4-cube	File:4-cube t0.svg	hypercube n-cube (Bn family)	{4,3,3} Template:CDD	16	32	24 {4}	8 {4,3}	{3,3}	16-cell
24-cell icositetrachoron octaplex polyoctahedron (pO)	File:24-cell t0 F4.svg	Fn family	{3,4,3} Template:CDD	24	96	96 {3}	24 {3,4}	{4,3}	self-dual	F4 [3,4,3]	1152
600-cell hexacosichoron tetraplex polytetrahedron (pT)	File:600-cell graph H4.svg	n-pentagonal polytope (Hn family)	{3,3,5} Template:CDD	120	720	1200 {3}	600 {3,3}	{3,5}	120-cell	H4 [5,3,3]	14400
120-cell hecatonicosachoron dodecacontachoron dodecaplex polydodecahedron (pD)	File:120-cell graph H4.svg	n-pentagonal polytope (Hn family)	{5,3,3} Template:CDD	600	1200	720 {5}	120 {5,3}	{3,3}	600-cell

John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).

Norman Johnson advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").

The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula:

$N_0-N_1+N_2-N_3=0$

where N_k denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.

As configurations

A regular 4-polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices in each edge (each edge has 2 vertices), and 2 cells meet at each face (each face belongs to 2 cells), in any regular 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.

5-cell {3,3,3}	16-cell {3,3,4}	8-cell {4,3,3}	24-cell {3,4,3}	600-cell {3,3,5}	120-cell {5,3,3}
$\left[\begin{array}{c}\begin{array}{cccc}5& 4& 6& 4\\ 2& 10& 3& 3\\ 3& 3& 10& 2\\ 4& 6& 4& 5\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cccc}8& 6& 12& 8\\ 2& 24& 4& 4\\ 3& 3& 32& 2\\ 4& 6& 4& 16\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cccc}16& 4& 6& 4\\ 2& 32& 3& 3\\ 4& 4& 24& 2\\ 8& 12& 6& 8\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cccc}24& 8& 12& 6\\ 2& 96& 3& 3\\ 3& 3& 96& 2\\ 6& 12& 8& 24\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cccc}120& 12& 30& 20\\ 2& 720& 5& 5\\ 3& 3& 1200& 2\\ 4& 6& 4& 600\end{array}\end{array}\right]$	$\left[\begin{array}{c}\begin{array}{cccc}600& 4& 6& 4\\ 2& 1200& 3& 3\\ 5& 5& 720& 2\\ 20& 30& 12& 120\end{array}\end{array}\right]$

Visualization

The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.

A4 = [3,3,3]	B4 = [4,3,3]		F4 = [3,4,3]	H4 = [5,3,3]
5-cell	16-cell	8-cell	24-cell	600-cell	120-cell
{3,3,3}	{3,3,4}	{4,3,3}	{3,4,3}	{3,3,5}	{5,3,3}
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Solid 3D orthographic projections
File:Tetrahedron.png Tetrahedral envelope (cell/vertex-centered)	File:16-cell ortho cell-centered.png Cubic envelope (cell-centered)	File:Hexahedron.png Cubic envelope (cell-centered)	File:Ortho solid 24-cell.png Cuboctahedral envelope (cell-centered)	File:Ortho solid 600-cell.png Pentakis icosidodecahedral envelope (vertex-centered)	File:Ortho solid 120-cell.png Truncated rhombic triacontahedron envelope (cell-centered)
Wireframe Schlegel diagrams (Perspective projection)
File:Schlegel wireframe 5-cell.png Cell-centered	File:Schlegel wireframe 16-cell.png Cell-centered	File:Schlegel wireframe 8-cell.png Cell-centered	File:Schlegel wireframe 24-cell.png Cell-centered	File:Schlegel wireframe 600-cell vertex-centered.png Vertex-centered	File:Schlegel wireframe 120-cell.png Cell-centered
Wireframe stereographic projections (3-sphere)
File:Stereographic polytope 5cell.png	File:Stereographic polytope 16cell.png	File:Stereographic polytope 8cell.png	File:Stereographic polytope 24cell.png	File:Stereographic polytope 600cell.png	File:Stereographic polytope 120cell.png

Regular star (Schläfli–Hess) 4-polytopes

File:Relationship among regular star polychora.png

This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.

File:Relationship among regular star polychora-8.png

A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations {a,g,s} are commutable, defining a cubic framework. There are 7 densities seen in vertical positioning, with 2 dual forms having the same density.

The Schläfli–Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes). They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is [[pentagram|Template:Sfrac]]. They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra, which are in turn analogous to the pentagram.

Names

Their names given here were given by John Conway, extending Cayley's names for the Kepler–Poinsot polyhedra: along with stellated and great, he adds a grand modifier. Conway offered these operational definitions:

1. stellation – replaces edges with longer edges in same lines. (Example: a pentagon stellates into a pentagram)
2. greatening – replaces the faces with large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
3. aggrandizement – replaces the cells with large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell)

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicosahedron {3,5,Template:Sfrac} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: g, a, and s for great, (ag)grand, and stellated. The final stellation, the great grand stellated polydodecahedron contains them all as gaspD.

Symmetry

All ten polychora have [3,3,5] (H₄) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].

Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

Properties

Note:

• There are 2 unique vertex arrangements, matching those of the 120-cell and 600-cell.
• There are 4 unique edge arrangements, which are shown as wireframes orthographic projections.
• There are 7 unique face arrangements, shown as solids (face-colored) orthographic projections.

The cells (polyhedra), their faces (polygons), the polygonal edge figures and polyhedral vertex figures are identified by their Schläfli symbols.

Name Conway (abbrev.)	Orthogonal projection	Schläfli Coxeter	C {p, q}	F {p}	E {r}	V {q, r}	Dens.	χ
Icosahedral 120-cell polyicosahedron (pI)	File:Ortho solid 007-uniform polychoron 35p-t0.png	{3,5,5/2} Template:CDD	120 {3,5} File:Icosahedron.png	1200 {3} File:Regular triangle.svg	720 {5/2} File:Star polygon 5-2.svg	120 {5,5/2} File:Great dodecahedron.png	4	480
Small stellated 120-cell stellated polydodecahedron (spD)	File:Ortho solid 010-uniform polychoron p53-t0.png	{5/2,5,3} Template:CDD	120 {5/2,5} File:Small stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	1200 {3} File:Regular triangle.svg	120 {5,3} File:Dodecahedron.png	4	−480
Great 120-cell great polydodecahedron (gpD)	File:Ortho solid 008-uniform polychoron 5p5-t0.png	{5,5/2,5} Template:CDD	120 {5,5/2} File:Great dodecahedron.png	720 {5} File:Regular pentagon.svg	720 {5} File:Regular pentagon.svg	120 {5/2,5} File:Small stellated dodecahedron.png	6	0
Grand 120-cell grand polydodecahedron (apD)	File:Ortho solid 009-uniform polychoron 53p-t0.png	{5,3,5/2} Template:CDD	120 {5,3} File:Dodecahedron.png	720 {5} File:Regular pentagon.svg	720 {5/2} File:Star polygon 5-2.svg	120 {3,5/2} File:Great icosahedron.png	20	0
Great stellated 120-cell great stellated polydodecahedron (gspD)	File:Ortho solid 012-uniform polychoron p35-t0.png	{5/2,3,5} Template:CDD	120 {5/2,3} File:Great stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	720 {5} File:Regular pentagon.svg	120 {3,5} File:Icosahedron.png	20	0
Grand stellated 120-cell grand stellated polydodecahedron (aspD)	File:Ortho solid 013-uniform polychoron p5p-t0.png	{5/2,5,5/2} Template:CDD	120 {5/2,5} File:Small stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	720 {5/2} File:Star polygon 5-2.svg	120 {5,5/2} File:Great dodecahedron.png	66	0
Great grand 120-cell great grand polydodecahedron (gapD)	File:Ortho solid 011-uniform polychoron 53p-t0.png	{5,5/2,3} Template:CDD	120 {5,5/2} File:Great dodecahedron.png	720 {5} File:Regular pentagon.svg	1200 {3} File:Regular triangle.svg	120 {5/2,3} File:Great stellated dodecahedron.png	76	−480
Great icosahedral 120-cell great polyicosahedron (gpI)	File:Ortho solid 014-uniform polychoron 3p5-t0.png	{3,5/2,5} Template:CDD	120 {3,5/2} File:Great icosahedron.png	1200 {3} File:Regular triangle.svg	720 {5} File:Regular pentagon.svg	120 {5/2,5} File:Small stellated dodecahedron.png	76	480
Grand 600-cell grand polytetrahedron (apT)	File:Ortho solid 015-uniform polychoron 33p-t0.png	{3,3,5/2} Template:CDD	600 {3,3} File:Tetrahedron.png	1200 {3} File:Regular triangle.svg	720 {5/2} File:Star polygon 5-2.svg	120 {3,5/2} File:Great icosahedron.png	191	0
Great grand stellated 120-cell great grand stellated polydodecahedron (gaspD)	File:Ortho solid 016-uniform polychoron p33-t0.png	{5/2,3,3} Template:CDD	120 {5/2,3} File:Great stellated dodecahedron.png	720 {5/2} File:Star polygon 5-2.svg	1200 {3} File:Regular triangle.svg	600 {3,3} File:Tetrahedron.png	191	0

See also

• Regular polytope
• List of regular polytopes
• Infinite regular 4-polytopes:
  • One regular Euclidean honeycomb: {4,3,4}
  • Four compact regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
  • Eleven paracompact regular hyperbolic honeycombs: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
• Abstract regular 4-polytopes:
  • 11-cell {3,5,3}
  • 57-cell {5,3,5}
• Uniform 4-polytope uniform 4-polytope families constructed from these 6 regular forms.
• Platonic solid
• Kepler-Poinsot polyhedra — regular star polyhedron
• Star polygon — regular star polygons
• 4-polytope
• 5-polytope
• 6-polytope

References

Bibliography

• • (Paper 10)

External links

• Jonathan Bowers, 16 regular 4-polytopes
• Regular 4D Polytope Foldouts
• Catalog of Polytope Images A collection of stereographic projections of 4-polytopes.
• A Catalog of Uniform Polytopes
• Dimensions 2 hour film about the fourth dimension (contains stereographic projections of all regular 4-polytopes)
• Reguläre Polytope
• The Regular Star Polychora
• Hypersolids