Simplex

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In Geometry , a simplex (plural ''simplexes'' or ''simplices'') or '''''n''-simplex''' is an ''n''-dimensional analogue of a triangle. Specifically, a simplex is the Convex Hull of a set of (''n'' + 1) Affinely Independent Point s in some Euclidean Space of dimension ''n'' or higher (i.e., a set of points such that no ''m''- Plane contains more than (''m'' + 1) of them; such points are said to be in General Position ).

For example, a 0-simplex is a Point , a 1-simplex is a Line Segment , a 2-simplex is a Triangle , a 3-simplex is a Tetrahedron , and a 4-simplex is a Pentachoron (in each case with interior).

A regular simplex is a simplex that is also a Regular Polytope . A regular ''n''-simplex may be constructed from a regular (''n'' − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.

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 ''C''(''n'' + 1, ''m'' + 1). 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 .

The regular simplex family is the first of three Regular Polytope families, labeled by Coxeter as ''α_n'', the other two being the Cross-polytope family, labeled as ''β_n'', and the Hypercube s, labeled as ''γ_n''. A fourth family, the Infinite Tessellation Of Hypercubes he labeled as ''δ_n''.

THE STANDARD SIMPLEX

The standard ''n''-simplex is the subset of '''R'''^''n''+1 given by
:

\Delta^n = \left\{(t_0,\cdots,t_n)\in\mathbb{R}^{n+1}\mid\Sigma_{i}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i
ight\}

The simplex Δ^''n'' live in the Affine Hyperplane obtained by removing the restriction ''t''_''i'' ≥ 0 in the above definition. The standard simplex is clearly regular.

The vertices of the standard ''n''-simplex are the points

e

dots

e

There is a canonical map from the standard ''n''-simplex to an arbitrary ''n''-simplex with vertices (''v''₀, …, ''v''_''n'') given by
:

(t_0,\cdots,t_n) \mapsto \Sigma_i t_i v_i

The coefficients ''t''_''i'' are called the Barycentric Coordinates of a point in the ''n''-simplex. Such a general simplex is often called an affine ''n''-simplex, to emphasize that the canonical map is an Affine Transformation . It is also sometimes called an '''oriented affine ''n''-simplex''' to emphasize that the canonical map may be Orientation Preserving or reversing.

GEOMETRIC PROPERTIES

The oriented Volume of an ''n''-simplex in ''n''-dimensional space with vertices (''v''₀, ..., ''v''_''n'') is

:

{1\over n!}\det
 \begin{pmatrix}
  v_0-v_1 & v_1-v_2& \dots & v_{n-1}-v_{n}
 \end{pmatrix}

where each column of the ''n'' × ''n'' Determinant is the difference between two vertices. Any determinant which involves taking the difference between pairs of vertices, where the pairs connect the vertices as a simply connected graph will also give the (same) volume. Without the 1/''n''! it is the formula for the volume of an ''n''- Parallelepiped . One way to understand the 1/''n''! factor is as follows. If the coordinates of a point in a unit ''n''-box are sorted, together with 0 and 1, and successive differences are taken, then since the results add to one, the result is a point in an ''n'' simplex spanned by the origin and the closest ''n'' vertices of the box. The taking of differences was an orthogonal (volume-preserving) transformation, but sorting compressed the space by a factor of ''n''!.

The Volume under a standard ''n''-simplex (i.e. between the origin and the simplex) is

:

{1 \over (n+1)!}

The Volume of a regular ''n''-simplex with unit side length is

:

{rac{\sqrt{n+1}}{n!\sqrt{2^n}}}

as can be seen by multiplying the previous formula by ''x''^''n+1'', to get the volume under the ''n''-simplex as a function of its vertex distance ''x'' from the origin, differentiating with respect to ''x'', at

x=1/\sqrt{2}

(where the ''n''-simplex side length is 1), and normalizing by the length

dx/\sqrt{n+1}\,

of the increment,

(dx/(n+1),\dots, dx/(n+1))

, along the normal vector.

Simplexes with an "orthogonal corner"

Orthogonal corner means here, that there is a vertex at which all adjacent hyperfaces are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists an n-dimensional version of the Pythagorean Theorem :

The sum of the squared n-dimensional volumes of the hyperfaces adjacent to the orthogonal corner equals the squared n-dimensional volume of the hyperface opposite of the orthogonal corner.

	CATEGORIES ABOUT SIMPLEX

	polytopes

	topology

	multi-dimensional geometry

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Simplex