# Convex regular polychoron

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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
Coxeter
Vertices Edges Faces Cells Vertex
figures
Dual Symmetry group
5-cell
pentachoron
pentatope
4-simplex
simplex
(n-simplex)
{3,3,3}
5 10 10
{3}
5
{3,3}
{3,3} (self-dual) A4
[3,3,3]
120
8-cell
octachoron
tesseract
4-cube
hypercube
(n-cube)
{4,3,3}
16 32 24
{4}
8
{4,3}
{3,3} 16-cell B4
[4,3,3]
384
16-cell
hexadecachoron
4-orthoplex
cross-polytope
(n-orthoplex)
{3,3,4}
8 24 32
{3}
16
{3,3}
{3,4} 8-cell B4
[4,3,3]
384
24-cell
icositetrachoron
octaplex
polyoctahedron
{3,4,3}
24 96 96
{3}
24
{3,4}
{4,3} (self-dual) F4
[3,4,3]
1152
120-cell
hecatonicosachoron
dodecaplex
polydodecahedron
dodecahedral
pentagonal polytope
(n-pentagonal polytope)
{5,3,3}
600 1200 720
{5}
120
{5,3}
{3,3} 600-cell H4
[5,3,3]
14400
600-cell
hexacosichoron
tetraplex
polytetrahedron
icosahedral
pentagonal polytope
(n-pentagonal polytope)
{3,3,5}
120 720 1200
{3}
600
{3,3}
{3,5} 120-cell H4
[5,3,3]
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 Nk 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}
Wireframe orthographic projections inside Petrie polygons.
Solid orthographic projections

tetrahedral
envelope

(cell/vertex-centered)

cubic envelope
(cell-centered)

Cubic envelope
(cell-centered)

cuboctahedral
envelope

(cell-centered)

truncated rhombic
triacontahedron
envelope

(cell-centered)

Pentakis icosidodecahedral
envelope

(vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)

(Vertex-centered)

(Cell-centered)

(Cell-centered)

(Cell-centered)

(Cell-centered)

(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)