Convex regular 4-polytope

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In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which 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 polytopes 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 three-dimensional equivalent.

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 in a regular fashion.

1 Properties
2 Visualizations
3 See also
4 References
5 External links

[edit] 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.

Name	Family	Schläfli symbol	Vertices	Edges	Faces	Cells	Vertex figures	Dual polytope	Symmetry group
pentachoron	simplex	{3,3,3}	5	10	10 triangles	5 tetrahedra	tetrahedra	(self-dual)	A₄	120
tesseract	hypercube	{4,3,3}	16	32	24 squares	8 cubes	tetrahedra	16-cell	B₄	384
16-cell	cross-polytope	{3,3,4}	8	24	32 triangles	16 tetrahedra	octahedra	tesseract	B₄	384
24-cell		{3,4,3}	24	96	96 triangles	24 octahedra	cubes	(self-dual)	F₄	1152
120-cell		{5,3,3}	600	1200	720 pentagons	120 dodecahedra	tetrahedra	600-cell	H₄	14400
600-cell		{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.).

[edit] Visualizations

The following table shows some 2 dimensional projections of these polytopes. Various other visualizations can be found in the external links below.

{3,3,3}	{4,3,3}	{3,3,4}	{3,4,3}	{5,3,3}	{3,3,5}
Wireframe orthographic projections

Solid orthographic projections (cell-centered)
tetrahedral envelope	cubic envelope	octahedral envelope	cuboctahedral envelope	truncated rhombic triacontahedron envelope	pentakis dodecahedral envelope
Wireframe Schlegel diagrams (Perspective projection)
(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)

[edit] See also

Infinite regular 4-polytopes:
- 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 4-polytopes:
- Schläfli-Hess polychoron - Ten nonconvex regular 4-polytopes
Abstract regular polychora:
- 11-cell {3,5,3}
- 57-cell {5,3,5}
Uniform polychoron 4-polytope families constructed from the from these 6 regular forms.
Regular polytope
Platonic solid

[edit] 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