Definition:Simplex

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Definition

A simplex is an $n$-dimensional generalization of a triangle and tetrahedron, for $n \in \Z_{>0}$.

A $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices.


Definition 1

A simplex $S$ in $\R^n$ with vertices $\family {\alpha_i}_{i \mathop = 0}^n$ is a set such that:

$S = \set {\ds \sum_{i \mathop = 0}^n \theta_i \alpha_i}$

where:

$\sequence {\alpha_i}_{i \mathop = 0}^n$ is a sequence of $n + 1$ affinely independent points in $\R^n$
$\sequence {\theta_i}_{i \mathop = 0}^n$ is a sequence of arbitrary real numbers such that:
$\forall i \in \set {0, 1, 2, \ldots, n}: \theta_i \in \hointr 0 \to$
$\ds \sum_{i \mathop = 0}^n \theta_i = 1$


Definition 2

A simplex $S$ in $\R^n$ with vertices $\family {\alpha_i}_{i \mathop = 0}^n$ is a set such that:

$S = \set {\ds \sum_{i \mathop = 0}^n \theta_i \alpha_i}$

where:

$\sequence {\alpha_i}_{i \mathop = 0}^n$ is a sequence of $n + 1$ affinely independent points in $\R^n$
$\sequence {\theta_i}_{i \mathop = 0}^n$ is a sequence of arbitrary real numbers such that:
$\forall i \in \set {0, 1, 2, \ldots, n}: \theta_i \in \closedint 0 1$
$\ds \sum_{i \mathop = 0}^n \theta_i = 1$


Dimension of Simplex

Let $S$ be a $k$-simplex:

$S = \set {\ds \sum_{i \mathop = 0}^k \theta_i \alpha_i}$

The parameter $k$ is called the dimension of $S$.


Examples

Tetrahedron

The tetrahedron is an example of a simplex of $3$ dimensions.


Also see

  • Results about simplices can be found here.


Linguistic Note

The plural of simplex has one of two possible forms:

simplexes, pronounced sim-plex-iz
simplices, pronounced sim-pli-seez

Either can be found on $\mathsf{Pr} \infty \mathsf{fWiki}$, depending on where the material was sourced from.

Compare the plural forms of vertex: vertices and index: indices.


Sources

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