Convex regular 4-polytope

In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a 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 where 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.

Name Family Schläfli
symbol 		Vertices Edges Faces Cells Vertex figures Dual polytope
pentachoron simplex {3,3,3} 5 10 10
triangles 		5
tetrahedra 		tetrahedra (self-dual)
tesseract measure polytope {4,3,3} 16 32 24
squares 		8
cubes 		tetrahedra 16-cell
16-cell cross-polytope {3,3,4} 8 24 32
triangles 		16
tetrahedra 		octahedra tesseract
24-cell {3,4,3} 24 96 96
triangles 		24
octahedra 		cubes (self-dual)
120-cell {5,3,3} 600 1200 720
pentagons 		120
dodecahedra 		tetrahedra 600-cell
600-cell {3,3,5} 120 720 1200
triangles 		600
tetrahedra 		icosahedra 120-cell

Note that since 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:

N0 − N1 + N2 − N3 = 0

where Nk denotes the number of k-faces in the polytope (a vertex is a 0-face, a edge is a 1-face, etc.).

See also

References

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}

See also: Convex regular 4-polytope, 120-cell, 16-cell, 24-cell, 3-sphere, 600-cell, Cell (mathematics), Convex, Cross-polytope