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 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.
Examples
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, 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)