---
tag: YVJ15PGKDM
kind: concept
name: Simplicial polytope
msc: [52B05, 52B11, 52B12]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Simplicial_polytope
same_as:
  wikidata: Q7520906
  wikipedia: Simplicial polytope
---

## Summary

A simplicial polytope is a [[convex polytope|300K6CB1KH]] all of whose [[facets|DPDXN2FJC4]] are simplices.
Its boundary is then a [[simplicial complex|V8GZ0W1ZZ5]] homeomorphic to a sphere.
Simplicial polytopes are the natural setting of the face-counting theory of polytopes, including the [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]].
Points in general position always have a simplicial convex hull, so "most" polytopes are simplicial.

## Explanations

### L1 human

A simplex is the simplest solid of its dimension: a segment, a triangle, a tetrahedron, and so on.
A polytope is simplicial when every face on its surface is a simplex.
In three dimensions this means every face is a triangle.
The octahedron and the icosahedron are simplicial; the cube is not, because its faces are squares.