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Simplex (n-dimensional simplex)

An n-simplex is the simplest convex polytope in n dimensions: the convex hull of n+1 affinely independent points. It is fundamental in geometry, topology, numerical methods and combinatorics.

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Overview

In geometry, a simplex is the generalization of a triangle to arbitrary dimension. An n-simplex is the convex hull of n+1 affinely independent points: that is, n+1 points that do not lie in any hyperplane of lower dimension. Simplexes (also spelled simpleces or simplices) provide the most basic building blocks for polyhedral and topological constructions. Special cases include the 0-simplex (a point), the 1-simplex (a line segment), the 2-simplex (a triangle) and the 3-simplex (a tetrahedron).

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Characteristics and combinatorics

Combinatorially, an n-simplex has n+1 vertices and a well defined collection of faces: for each k with 0 ≤ k ≤ n, the number of k-dimensional faces is the binomial coefficient C(n+1, k+1). Thus the number of edges is C(n+1,2), the number of facets (codimension‑1 faces) is n+1, and the simplex is self-dual as a polytope. Each point inside an n-simplex can be uniquely expressed by barycentric coordinates relative to its vertices, which are nonnegative weights summing to 1.

Algebraic and metric descriptions

There are several convenient coordinate models. The standard n-simplex is often taken as the set of points (x0, x1, ..., xn) in R^{n+1} with all xi ≥ 0 and sum xi = 1. Equivalently, any n-simplex may be described as the image of this standard simplex under an affine map. The volume (hypervolume) of a simplex can be computed from the determinant of edge vectors or by the Cayley–Menger determinant, which expresses squared volume in terms of squared edge lengths.

Regular simplex and constructions

A regular simplex is one in which all edges have equal length; it is a regular polytope and can be constructed by taking a regular (n−1)-simplex and adjoining a new vertex equidistant from all existing vertices. Regular simplices are unique up to isometry for a given edge length. The 4-dimensional regular simplex is sometimes called the pentachoron or 5-cell (pentachoron), and a regular simplex remains a simplex under duality. The process of adding a single vertex joined to every vertex of an existing simplex is a standard way to increase dimension by one and create an (n)-simplex from an (n−1)-simplex by connecting a new vertex to all existing vertices.

Simplexes are central in algebraic topology (as the cells of simplicial complexes), computational geometry (triangulations and mesh generation), and numerical methods such as the finite element method, where simplicial elements approximate domains. The term also appears in optimization: the simplex algorithm operates on vertices of convex polyhedra to solve linear programs. In topology and combinatorics, simplicial complexes built from simplices encode connectivity and incidence information in a combinatorial way.

Notable facts and distinctions

  • An n-simplex has exactly n+1 facets and is the simplest n-dimensional convex polytope.
  • Any set of n+1 affinely independent points defines an n-simplex; conversely, the vertices of an n-simplex are affinely independent.
  • Volume formulas can be obtained from determinants; the Cayley–Menger determinant provides a general expression in terms of edge lengths.
  • Regular simplices are regular polytopes in all dimensions and are often used as reference shapes in geometric constructions (regular polytope).

For basic intuition, think of the 0-simplex as a point, the 1-simplex as a segment, the 2-simplex as a triangle and the 3-simplex as the familiar tetrahedron. Higher-dimensional simplices extend these properties in a straightforward combinatorial and geometric way, and they remain a compact, widely applicable concept across mathematics and its applications.

A 3-simplex or tetrahedron
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} V. These points are called affine independent if for the scalars {\displaystyle t_{0},\ldots ,t_{k}\in \mathbb {R} }holds that it {\displaystyle t_{0}+\cdots +t_{k}=0}follows from {\displaystyle v_{0}t_{0}+\cdots +v_{k}t_{k}=0}with that {\displaystyle 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+1points 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 {\displaystyle 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 spannedv_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{k}by the .

A set of points of an n-dimensional vector space Vover \mathbb {R} ( n\in \mathbb {N} ) is called in general position if every subset consisting of at most n+1points 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 k is the dimension of the simplex. A simplex of dimension k is also called for short k-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 d-sides (or d-facets) of the k-simplex is equal to the binomial coefficient {\tbinom {k+1}{d+1}}.

The n -simplex is the simplest n-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 n -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=1the volume {\displaystyle c^{n}/n!\,}.

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