Simplicial polytope

Not to be confused with Simple polytope.

In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces^[1] and corresponds via Steinitz's theorem to a maximal planar graph.

Pentagonal bipyramid

They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons.

Examples

Simplicial polyhedra include:

• Bipyramids
• Gyroelongated bipyramids
• Deltahedra (equilateral triangles)
  • Platonic
    • tetrahedron, octahedron, icosahedron
  • Johnson solids:
    • triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid
• Catalan solids:
  • triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron

Simplicial tilings:

• Regular:
  • triangular tiling
• Laves tilings:
  • tetrakis square tiling, triakis triangular tiling, kisrhombille 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

Notes

1. Polyhedra, Peter R. Cromwell, 1997. (p.341)

References

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