Simplicial polytope

Short description: Polytope whose facets are all simplices

Not to be confused with Simple polytope.

Pentagonal bipyramid

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.

## Contents

* 1 Examples
* 2 See also
* 3 Notes
* 4 References

## 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

* Simplicial complex
* Delaunay triangulation

## Notes

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

## References

* Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. ISBN 0-521-66405-5. https://books.google.com/books?id=OJowej1QWpoC&q=Polyhedra&pg=PP1.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Simplicial polytope. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Simplicial_polytope&oldid=4152607"