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.
A famous result by Gil Kalai states that a simple polytope is completely determined by its 1-skeleton.
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 polychora:
In higher dimensions:
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