Simplex

Hypertetrahedron is a redirect to this article. For the also so called 4-dimensional special case see pentachoron.

In geometry, a simplex or n-simplex, occasionally also an n-dimensional hypertetrahedron, is a special n-dimensional polytope.

A simplex is the simplest form of a polytope. Each -dimensional simplex has n+1 vertices. One creates an -simplex from an (n-1) -simplex by adding an affine independent point (see below) and connecting all vertices of the lower dimensional simplex to this point in the form of a cone formation by stretches. Thus, with increasing dimension, the series point, stretch, triangle, tetrahedron is obtained. An -simplex $(n\in \mathbb {N} )$ is the continuation of this series on dimensions.

A 3-simplex or tetrahedron

Definitions

Affine independence

Let $k\in \mathbb {N}$ and let be $v_{0},\ldots ,v_{k}$ finitely many points of an -vector space $\mathbb {R}$ . These points are called affine independent if for the scalars $t_{0},\ldots ,t_{k}\in \mathbb {R}$ holds that it $t_{0}+\cdots +t_{k}=0$ follows from $v_{0}t_{0}+\cdots +v_{k}t_{k}=0$ with that $t_{0}=\cdots =t_{k}=0$ .

In other words, there is no (k-1) -dimensional affine subspace $V_{0}\subset V$ , in which the k+1 points lie. An equivalent formulation is: the set $\{v_{1}-v_{0},\ldots ,v_{k}-v_{0}\}$ is linearly independent. In this case, each of the points $v_{j}\ (j=0,1,\ldots ,k)$ independent of the remaining points $v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{k}$ affinely independent and equally independent of the affine subspace spanned $v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{k}$ by the .

A set of points of an -dimensional vector space over $\mathbb {R}$ ( $n\in \mathbb {N}$ ) is called in general position if every subset consisting of at most n+1 points is affinely independent.

Simplex

Let $k\in \mathbb {N}$ and let $v_{0},\ldots ,v_{k}$ affine independent points of $\mathbb {R} ^{n}$ (or an n-dimensional vector space over $\mathbb {R}$ ), then the simplex Δ $v_{0},\ldots ,v_{k}$ spanned (or generated) by $\Delta$ is equal to the following set:

$\Delta =\left\{x\in \mathbb {R} ^{n}:x=\sum _{i=0}^{k}t_{i}v_{i}\ {\text{mit}}\ 0\leq t_{i}\leq 1\ {\text{und}}\ \sum _{i=0}^{k}t_{i}=1\right\}$ .

The points $v_{i}$ are called vertices of Δ $\Delta$ and $(t_{0},...,t_{k})\in [0,1]^{k+1}$ barycentric coordinates. The number is the dimension of the simplex. A simplex of dimension is also called for short -simplex. Thus, a simplex is nothing more than the convex hull of finitely many affine independent points in $\mathbb {R} ^{n}$ , which are then the vertices of this simplex.

Sides and edge

Let Δ be $\Delta$ a simplex. Any $\Delta$ simplex contained in Δ $\Delta$ spanned by a nonempty subset of the vertices of Δ is called a side (more rarely facet or sub-simplex) of Δ $\Delta$ . The zero-dimensional sides (facets) are just the vertices or corners, the 1-sides (or 1-facets) are the edges, and the (k-1) }-sides or (k-1) }-facets are called side faces. The union of the side faces is called the edge ∂ $\partial \Delta$ of the simplex Δ $\Delta$ :

$\partial \Delta =\left\{x\in \mathbb {R} ^{n}:x=\sum _{i=0}^{k}t_{i}v_{i}\ {\text{mit}}\ 0\leq t_{i}\leq 1\ {\text{und mindestens einem}}\ t_{i}=0,{\text{sowie}}\ \sum _{i=0}^{k}t_{i}=1\right\}$

The number of -sides (or -facets) of the -simplex is equal to the binomial coefficient ${\tbinom {k+1}{d+1}}$ .

The -simplex is the simplest -dimensional polytope, measured by the number of vertices. The simplex method from linear optimization is named after the simplex, and so is the downhill simplex method in nonlinear optimization.

Example

A 0 simplex is a point.
A 1-simplex is a distance.
A 2-simplex is a triangle.
A 3-simplex is a tetrahedron (four corners, four sides of triangles, six edges); it is created from a triangle (2-simplex) to which a point not in the triangle plane is added and connected to all corners of the triangle.
A 4-simplex is also called a pentachoron.
An example of an -simplex in $\mathbb {R} ^{n}$ (and specifically one with a right-angled corner at the origin) is given by

$\left\{x\in \mathbb {R} ^{n}\mid x_{i}\geq 0,\,\sum \limits _{i=1}^{n}x_{i}\leq 1\right\}$

given. This simplex is called a unit simplex. It is spanned by the zero vector and the unit vectors $e_{1},\dotsc ,e_{n}$ of the standard basis of $\mathbb {R} ^{n}$ spanned by the length of the unit vectors has c=1 the volume $c^{n}/n!\,$ .

Simplex

Definitions

Affine independence

Simplex

Sides and edge

Example

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