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[edit]

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}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8 24 32
{3}
16
{3,3}
{3,4} 8-cell B4
[4,3,3]
384
24-cell
icositetrachoron
octaplex
polyoctahedron
{3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
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}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
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[edit]

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}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Wireframe orthographic projections inside Petrie polygons.
4-simplex t0.svg 4-cube t0.svg 4-cube t3.svg 24-cell t0 F4.svg 120-cell graph H4.svg 600-cell graph H4.svg
Solid orthographic projections
Tetrahedron.png
tetrahedral
envelope

(cell/vertex-centered)
Hexahedron.png
cubic envelope
(cell-centered)
16-cell ortho cell-centered.png
Cubic envelope
(cell-centered)
Ortho solid 24-cell.png
cuboctahedral
envelope

(cell-centered)
Ortho solid 120-cell.png
truncated rhombic
triacontahedron
envelope

(cell-centered)
Ortho solid 600-cell.png
Pentakis icosidodecahedral
envelope

(vertex-centered)
Wireframe Schlegel diagrams (Perspective projection)
Schlegel wireframe 5-cell.png
(Vertex-centered)
Schlegel wireframe 8-cell.png
(Cell-centered)
Schlegel wireframe 16-cell.png
(Cell-centered)
Schlegel wireframe 24-cell.png
(Cell-centered)
Schlegel wireframe 120-cell.png
(Cell-centered)
Schlegel wireframe 600-cell vertex-centered.png
(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)
Stereographic polytope 5cell.png Stereographic polytope 8cell.png Stereographic polytope 16cell.png Stereographic polytope 24cell.png Stereographic polytope 120cell.png Stereographic polytope 600cell.png

See also[edit]

References[edit]

External links[edit]