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]
Micha Perles conjectured that a simple polytope is completely determined by its 1-skeleton; his conjecture was proven in 1987 by Blind and Mani-Levitska.[3] Gil Kalai provided a later simplification of this result based on the theory of unique sink orientations.[4]
Examples
In three dimensions:
- Prisms
- Platonic solids:
- Archimedean solids:
- Goldberg polyhedron and Fullerenes:
- In general, any polyhedron can be made into a simple one by truncating its vertices of valence 4 or higher.
In four dimensions:
- Regular:
- 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
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.