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]

A famous result by Gil Kalai states that a simple polytope is completely determined by its 1-skeleton.


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)