Simple polytope

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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]


In three dimensions:

In four dimensions:

In higher dimensions:

See also[edit]


  1. ^ Lectures on Polytopes, by Günter M. Ziegler (1995) ISBN 0-387-94365-X
  2. ^ Polyhedra, Peter R. Cromwell, 1997. (p.341)
  3. ^ Blind, Roswitha; Mani-Levitska, Peter (1987), "Puzzles and polytope isomorphisms", Aequationes Mathematicae, 34 (2-3): 287–297, doi:10.1007/BF01830678, MR 921106 .
  4. ^ 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 .