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.

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.

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

Examples

In three dimensions:

• Prisms
• Truncated trapezohedrons
• Platonic solids:
  • tetrahedron, cube, dodecahedron
• Archimedean solids:
  • truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, truncated icosidodecahedron
• In general, any truncated polyhedron is simple, including examples:
  • truncated rhombic dodecahedron
  • truncated rhombic triacontahedron

In four dimensions:

• Regular:
  • 120-cell, Tesseract
• Uniform polychorons:
  • truncated 5-cell, truncated tesseract, truncated 24-cell, truncated 120-cell
  • all bitruncated, cantitruncated or omnitruncated polychora
  • duoprisms

In higher dimensions:

• d-simplex
• hypercube
• associahedron
• permutohedron
• all omnitruncated polytopes

See also

• Dehn-Sommerville equations
• Voronoi tessellation

References

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