Simplicial 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.

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)

See also

Notes

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

References

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

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