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The tesseract is one of 6 convex regular polychora

In mathematics, a convex regular polychoron is a polychoron (4-polytope) that is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions).

These polychora were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no exact three-dimensional equivalent.

Each convex regular polychoron 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 in a regular fashion.

Properties

The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora 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	Family	Schläfli symbol	Vertices	Edges	Faces	Cells	Vertex figures	Dual polytope	Symmetry group
5-cell pentachoron pentatope hyperpyramid hypertetrahedron 4-simplex	simplex (n-simplex)	{3,3,3}	5	10	10 triangles	5 tetrahedra	tetrahedra	(self-dual)	A4	120
8-cell octachoron tesseract hypercube 4-cube	hypercube (n-cube)	{4,3,3}	16	32	24 squares	8 cubes	tetrahedra	16-cell	B4	384
16-cell hexadecachoron hyperoctahedron 4-orthoplex	cross-polytope (n-orthoplex)	{3,3,4}	8	24	32 triangles	16 tetrahedra	octahedra	8-cell	B4	384
24-cell icositetrachoron octaplex polyoctahedron		{3,4,3}	24	96	96 triangles	24 octahedra	cubes	(self-dual)	F4	1152
120-cell hecatonicosachoron dodecaplex hyperdodecahedron polydodecahedron	dodecahedral pentagonal polytope (n-pentagonal polytope)	{5,3,3}	600	1200	720 pentagons	120 dodecahedra	tetrahedra	600-cell	H4	14400
600-cell hexacosichoron tetraplex hypericosahedron polytetrahedron	icosahedral pentagonal polytope (n-pentagonal polytope)	{3,3,5}	120	720	1200 triangles	600 tetrahedra	icosahedra	120-cell	H4	14400

Since the boundaries of each of these figures is topologically equivalent to a 3-sphere, whose Euler characteristic is zero, we have the 4-dimensional analog 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.).

Visualizations

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

5-cell	Tesseract	16-cell	24-cell	120-cell	600-cell
{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}
Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Wireframe orthographic projections inside Petrie polygons.
		File:4-cube t3.svg	File:24-cell t0 F4.svg	File:120-cell graph H4.svg	File:600-cell graph H4.svg
Solid orthographic projections
File:Tetrahedron.png tetrahedral envelope (cell/vertex-centered)	File:Hexahedron.png cubic envelope (cell-centered)	File:16-cell ortho cell-centered.png Cubic envelope (cell-centered)	File:Ortho solid 24-cell.png cuboctahedral envelope (cell-centered)	File:Ortho solid 120-cell.png truncated rhombic triacontahedron envelope (cell-centered)	File:Ortho solid 600-cell.png Pentakis icosidodecahedral envelope (vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
File:Schlegel wireframe 5-cell.png (Vertex-centered)	File:Schlegel wireframe 8-cell.png (Cell-centered)	File:Schlegel wireframe 16-cell.png (Cell-centered)	File:Schlegel wireframe 24-cell.png (Cell-centered)	File:Schlegel wireframe 120-cell.png (Cell-centered)	File:Schlegel wireframe 600-cell vertex-centered.png (Vertex-centered)
Wireframe stereographic projections (Hyperspherical)
File:Stereographic polytope 5cell.png	File:Stereographic polytope 8cell.png	File:Stereographic polytope 16cell.png	File:Stereographic polytope 24cell.png	File:Stereographic polytope 120cell.png	File:Stereographic polytope 600cell.png

See also

• Infinite regular polychora:
  • One regular Euclidean honeycomb: {4,3,4}
  • Four regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
• Nonconvex regular polychora:
  • Schläfli-Hess polychoron - Ten nonconvex regular polychora
• Abstract regular polychora:
  • 11-cell {3,5,3}
  • 57-cell {5,3,5}
• Uniform polychoron Polychoron families constructed from the from these 6 regular forms.
• Regular polytope
• Platonic solid

References

• H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. ISBN 0-471-50458-0.
• H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
• D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

External links

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• Jonathan Bowers, 16 regular polychora
• 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 polychorons)