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

[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 Nk 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.

Orthographic projections
Stereographic projection
(Schlegel diagrams)

(Cell-centered)

(Cell-centered)

(Cell-centered)

(Cell-centered)

(Cell-centered)

(Vertex-centered)
{3,3,3} {4,3,3} {3,3,4} {3,4,3} {5,3,3} {3,3,5}

[edit] See also

[edit] References

[edit] External links

Convex regular 4-polytopes
pentachoron 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}
In other languages