Elements
The convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m+1 (of the n+1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient . Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex. See Simplicial complex#Definitions
The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn.
The number of 1-faces (edges) of the n-simplex is the (n-1)th triangle number, the number of 2-faces (faces) of the n-simplex is the (n-2)th tetrahedron number, the number of 3-faces (cells) of the n-simplex is the (n-3)th pentachoron number, and so on.
| Δn | Name | Schläfli symbol Coxeter-Dynkin |
0- faces |
1- faces |
2- faces |
3- faces |
4- faces |
5- faces |
6- faces |
7- faces |
8- faces |
9- faces |
10- faces |
Sum =2n+1-1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Δ0 | 0-simplex (point) |
1 | 1 | |||||||||||
| Δ1 | 1-simplex (line segment) |
{} |
2 | 1 | 3 | |||||||||
| Δ2 | 2-simplex (triangle) |
{3} |
3 | 3 | 1 | 7 | ||||||||
| Δ3 | 3-simplex (tetrahedron) |
{3,3} |
4 | 6 | 4 | 1 | 15 | |||||||
| Δ4 | 4-simplex (5-cell) |
{3,3,3} |
5 | 10 | 10 | 5 | 1 | 31 | ||||||
| Δ5 | 5-simplex | {3,3,3,3} |
6 | 15 | 20 | 15 | 6 | 1 | 63 | |||||
| Δ6 | 6-simplex | {3,3,3,3,3} |
7 | 21 | 35 | 35 | 21 | 7 | 1 | 127 | ||||
| Δ7 | 7-simplex | {3,3,3,3,3,3} |
8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | 255 | |||
| Δ8 | 8-simplex | {3,3,3,3,3,3,3} |
9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | 511 | ||
| Δ9 | 9-simplex | {3,3,3,3,3,3,3,3} |
10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | 1023 | |
| Δ10 | 10-simplex | {3,3,3,3,3,3,3,3,3} |
11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | 2047 |
In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
Read more about this topic: Simplex
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