Simplicial polytope

Not to be confused with Simple polytope.

In geometry, a simplicial polytope is a polytope whose facets are all simplices. It is topologically dual to simple polytopes. Polytopes that are both simple and simplicial are either simplices or two-dimensional polygons.

Examples of simplicial polytopes

Pentagonal bipyramid, an example of a simplicial 3-tope

5-cell, an example of 4-tope

Triangular tiling

Examples

In the case of a three-dimensional simplicial polytope, known as the simplicial polyhedron, the polytope contains only triangular faces of any type.^[1] These polyhedra include bipyramids, gyroelongated bipyramids, deltahedra (wherein the faces are equilateral triangles, and Kleetope of polyhedra. The simplicial polyhedron corresponds via Steinitz's theorem to a maximal planar graph.

For a simplicial tiling, examples are triangular tiling and Laves tiling.

Simplicial 4-polytopes include:

• convex regular 4-polytope
  • 4-simplex, 16-cell, 600-cell
• Dual convex uniform honeycombs:
  • Disphenoid tetrahedral honeycomb
  • Dual of cantitruncated cubic honeycomb
  • Dual of omnitruncated cubic honeycomb
  • Dual of cantitruncated alternated cubic honeycomb

Simplicial higher polytope families:

• simplex
• cross-polytope (Orthoplex)

See also

• Simplicial complex
• Delaunay triangulation