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.
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.
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. Gil Kalai provided a later simplification of this result based on the theory of unique sink orientations.
In three dimensions:
- 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:
- Uniform 4-polytope:
In higher dimensions:
- 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.
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