Simplex - Elements

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.

n-Simplex elements
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

Famous quotes containing the word elements:

    Kitsch is the daily art of our time, as the vase or the hymn was for earlier generations. For the sensibility it has that arbitrariness and importance which works take on when they are no longer noticeable elements of the environment. In America kitsch is Nature. The Rocky Mountains have resembled fake art for a century.
    Harold Rosenberg (1906–1978)

    Let us have a fair field! This is all we ask, and we will be content with nothing less. The finger of evolution, which touches everything, is laid tenderly upon women. They have on their side all the elements of progress, and its spirit stirs within them. They are fighting, not for themselves alone, but for the future of humanity. Let them have a fair field!
    Tennessee Claflin (1846–1923)

    The elements of success in this business do not differ from the elements of success in any other. Competition is keen and bitter. Advertising is as large an element as in any other business, and since the usual avenues of successful exploitation are closed to the profession, the adage that the best advertisement is a pleased customer is doubly true for this business.
    Madeleine [Blair], U.S. prostitute and “madam.” Madeleine, ch. 5 (1919)