simplicial polytope
{{Short description|Polytope whose facets are all simplices}}
{{distinguish|Simple polytope}}
File:Pentagonale_bipiramide.png
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[https://books.google.com/books?id=OJowej1QWpoC Polyhedra, Peter R. Cromwell, 1997]. (p.341) 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 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
Notes
{{reflist}}
References
- {{cite book
| last = Cromwell
| first = Peter R.
| title = Polyhedra
| publisher = Cambridge University Press
| date = 1997
| isbn = 0-521-66405-5
| url=https://books.google.com/books?id=OJowej1QWpoC&q=Polyhedra&pg=PP1}}