In geometry, a simplicial polytope is a polytope whose facets are all simplices.
For example, a simplicial polyhedron in 3 dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph.
They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons.
Simplicial polyhedra include:
Bipyramids
Gyroelongated dipyramids
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, bisected hexagonal 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)
