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:
  • tetrahedron, cube, dodecahedron
• Archimedean solids:
  • truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, truncated icosidodecahedron
• Goldberg polyhedron and Fullerenes:
  • chamfered tetrahedron, chamfered cube, chamfered dodecahedron ...
• In general, any polyhedron can be made into a simple one by truncating its vertices of valence 4 or higher.
  • truncated trapezohedrons

In four dimensions:

• Regular:
  • 120-cell, Tesseract
• 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

• Dehn–Sommerville equations
• Voronoi tessellation

Notes

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.

References

• Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 0-521-66405-5.