Definition:Simplex

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:

$\forall i \in \set {0, 1, 2, \ldots, n}: \theta_i \in \closedint 0 1$

$\ds \sum_{i \mathop = 0}^n \theta_i = 1$

Also see

• Equivalence of Definitions of Simplex

• Definition:Pentatope

• Results about simplices can be found here.

Linguistic Note

The plural of simplex is simplices, pronounced sim-pli-seez.

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

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$

• Weisstein, Eric W. "Simplex." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Simplex.html