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.

Contents

1 Properties
2 See also
3 References
4 External links

Properties

The following table 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 5 10 10
triangles 5
tetrahedra tetrahedra (self-dual) A₄ 120
tesseract measure polytope 16 32 24
squares 8
cubes tetrahedra 16-cell B₄ 384
16-cell cross-polytope 8 24 32
triangles 16
tetrahedra octahedra tesseract B₄ 384
24-cell 24 96 96
triangles 24
octahedra cubes (self-dual) F₄ 1152
120-cell 600 1200 720
pentagons 120
dodecahedra tetrahedra 600-cell H₄ 14400
600-cell 120 720 1200
triangles 600
tetrahedra icosahedra 120-cell H₄ 14400

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

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

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.

External links

[Regular 4D Polytope Foldouts]
[Hyperspace Tutorial], various visualizations of the regular 4-polytopes

Convex regular 4-polytopes
pentachoron	tesseract	16-cell	24-cell	120-cell	600-cell

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Convex regular 4-polytope

Properties

See also

References

External links