Simplicial polytope

In geometry, a simplicial polytope is a d-polytope whose facets are all simplices.

For example, a simplicial polyhedron 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.

Contents

1 Examples
2 Notes
3 References
4 See also

Examples

Simplicial polyhedra include:

Bipyramids
Gyroelongated dipyramids
Deltahedra (Equilateral triangles)
- Platonic
  - tetrahedron, octahedron, icosahedron
- Johnson solids:
  - triangular dipyramid, gyroelongated square dipyramid, triaugmented triangular prism, snub disphenoid
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:
1. Disphenoid tetrahedral honeycomb
2. Dual of cantitruncated cubic honeycomb
3. Dual of omnitruncated cubic honeycomb
4. Dual of cantitruncated alternated cubic honeycomb

Simplicial higher polytope families:

simplex
cross-polytope (Orthoplex)

Notes

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

References

Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 0521664055. http://books.google.com/books?id=OJowej1QWpoC&lpg=PP1&dq=Polyhedra&pg=PP1#v=onepage&q=&f=false.

See also