Simple polytope

In geometry, a d-dimensional simple polytope is a d-dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets). The vertex figure of a simple d-polytope is a (d-1)-simplex.[1]

They are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons.

For example, a simple polyhedron is a polyhedron whose vertices are adjacent to 3 edges and 3 faces. And the dual to a simple polyhedron is a simplicial polyhedron, containing all triangular faces.[2]

A famous result by Blind, Mani-Levitska, and Kalai states that a simple polytope is completely determined by its 1-skeleton.[3][4]

Examples

In three dimensions:

Prisms
Platonic solids:
tetrahedron, cube, dodecahedron
Archimedean solids:
truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, truncated icosidodecahedron
Goldberg polyhedron and Fullerenes:
chamfered tetrahedron, chamfered cube, chamfered dodecahedron ...
In general, any polyhedron can be made into a simple one by truncating its vertices of valence 4 or higher.
truncated trapezohedrons

In four dimensions:

Regular:
120-cell, Tesseract
Uniform 4-polytope:
truncated 5-cell, truncated tesseract, truncated 24-cell, truncated 120-cell
all bitruncated, cantitruncated or omnitruncated 4-polytopes
duoprisms

In higher dimensions:

d-simplex
hypercube
associahedron
permutohedron
all omnitruncated polytopes

See also

Dehn-Sommerville equations
Voronoi tessellation

Notes

Lectures on Polytopes, by Günter M. Ziegler (1995) ISBN 0-387-94365-X
Polyhedra, Peter R. Cromwell, 1997. (p.341)
Blind, Roswitha; Mani-Levitska, Peter (1987), "Puzzles and polytope isomorphisms", Aequationes Mathematicae 34 (2-3): 287–297, doi:10.1007/BF01830678, MR 921106.

Kalai, Gil (1988), "A simple way to tell a simple polytope from its graph", Journal of Combinatorial Theory, Series A 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 964396.

References

Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 0-521-66405-5.

Mathematics Encyclopedia