Simplex


In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
For example,
Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.
More formally, suppose the k + 1 points are affinely independent, which means are linearly independent.
Then, the simplex determined by them is the set of points
A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex is the simplex formed from the k + 1 standard unit vectors, or
In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices.

History

The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines".
Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra".
In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum and then with the same Latin adjective in the normal form simplex.
The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the tessellation of n-dimensional space by infinitely many hypercubes, he labeled as δn.

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 is an m-simplex, called an m-face of the n-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -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. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail.
The number of 1-faces of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the th tetrahedron number, the number of 3-faces of the n-simplex is the th 5-cell number, and so on.
ΔnNameSchläfli
Coxeter		0-
faces
1-
faces
2-
faces
3-
faces
4-
faces
5-
faces
6-
faces
7-
faces
8-
faces
9-
faces
10-
faces
Sum
= 2n+1 − 1
Δ00-simplex

1 1
Δ11-simplex
 = ∨ = 2 ·
21 3
Δ22-simplex
 = 3 ·
331 7
Δ33-simplex
 = 4 ·
4641 15
Δ44-simplex
 = 5 ·
5101051 31
Δ55-simplex = 6 ·
615201561 63
Δ66-simplex = 7 ·
72135352171 127
Δ77-simplex = 8 ·
8285670562881 255
Δ88-simplex = 9 ·
93684126126843691 511
Δ99-simplex = 10 ·
104512021025221012045101 1023
Δ1010-simplex = 11 ·
1155165330462462330165551112047

In layman's terms, an n-simplex is a simple shape that requires n dimensions. Consider a line segment AB as a "shape" in a 1-dimensional space. One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space. One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space. One can place a new point E somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space and adding a second point, which required the increase to 1-dimensional space.
More formally, an -simplex can be constructed as a join of an n-simplex and a point, . An -simplex can be constructed as a join of an m-simplex and an n-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ∨ = 2 · . A general 2-simplex is the join of three points: ∨ ∨ . An isosceles triangle is the join of a 1-simplex and a point: ∨ . An equilateral triangle is 3· or . A general 3-simplex is the join of 4 points: ∨ ∨ ∨ . A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: ∨ ∨ . A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.∨ or ∨. A regular tetrahedron is 4 · or and so on.
In some conventions, the empty set is defined to be a -simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology than to the study of polytopes.

Symmetric graphs of regular simplices

These Petrie polygons show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

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The standard simplex

The standard n-simplex is the subset of Rn+1 given by
The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition.
The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices given by
The coefficients ti 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.
More generally, there is a canonical map from the standard -simplex onto any polytope with n vertices, given by the same equation :
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex:
A commonly used function from Rn to the interior of the standard -simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.

Examples

An alternative coordinate system is given by taking the indefinite sum:
This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:
Geometrically, this is an n-dimensional subset of rather than of . The facets, which on the standard simplex correspond to one coordinate vanishing, here correspond to successive coordinates being equal, while the interior corresponds to the inequalities becoming strict.
A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a fundamental domain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into mostly disjoint simplices, showing that this simplex has volume Alternatively, the volume can be computed by an iterated integral, whose successive integrands are
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

Projection onto the standard simplex

Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given with possibly negative entries, the closest point on the simplex has coordinates
where is chosen such that
can be easily calculated from sorting.
The sorting approach takes complexity, which can be improved to complexity via median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the ball.

Corner of cube

Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.

Cartesian coordinates for regular ''n''-dimensional simplex in R''n''

The coordinates of the vertices of a regular n-dimensional simplex can be obtained from these two properties,
  1. For a regular simplex, the distances of its vertices to its center are equal.
  2. The angle subtended by any two vertices of an n-dimensional simplex through its center is
These can be used as follows. Let vectors represent the vertices of an n-simplex center the origin, all unit vectors so a distance 1 from the origin, satisfying the first property. The second property means the dot product between any pair of the vectors is. This can be used to calculate positions for them.
For example, in three dimensions the vectors are the vertices of a 3-simplex or tetrahedron. Write these as
Choose the first vector v0 to have all but the first component zero, so by the first property it must be and the vectors become
By the second property the dot product of v0 with all other vectors is -, so each of their x components must equal this, and the vectors become
Next choose v1 to have all but the first two elements zero. The second element is the only unknown. It can be calculated from the first property using the Pythagorean theorem, and so the second vector can be completed:
The second property can be used to calculate the remaining y components, by taking the dot product of v1 with each and solving to give
From which the z components can be calculated, using the Pythagorean theorem again to satisfy the first property, the two possible square roots giving the two results
This process can be carried out in any dimension, using n + 1 vectors, applying the first and second properties alternately to determine all the values.

Geometric properties

Volume

The volume of an n-simplex in n-dimensional space with vertices is
where each column of the n × n determinant is the difference between the vectors representing two vertices. A more symmetric way to write it is
Another common way of computing the volume of the simplex is via the Cayley–Menger determinant. It can also compute the volume of a simplex embedded in a higher-dimensional space, e.g., a triangle in.
Without the 1/n! it is the formula for the volume of an n-parallelotope.
This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis of.
Given a permutation of, call a list of vertices a n-path if
. The following assertions hold:
If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotop is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of to. As previously, this implies that the volume of a simplex coming from a n-path is:
Conversely, given an n-simplex of, it can be supposed that the vectors form a basis of. Considering the parallelotope constructed from and, one sees that the previous formula is valid for every simplex.
Finally, the formula at the beginning of this section is obtained by observing that
From this formula, it follows immediately that the volume under a standard n-simplex is
The volume of a regular n-simplex with unit side length is
as can be seen by multiplying the previous formula by xn+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 , and normalizing by the length of the increment,, along the normal vector.

Dihedral angles of the regular n-simplex

Any two -dimensional faces of a regular n-dimensional simplex are themselves regular ''-dimensional simplices, and they have the same dihedral angle of cos−1.
This can be seen by noting that the center of the standard simplex is, and the centers of its faces are coordinate permutations of. Then, by symmetry, the vector pointing from to is perpendicular to the faces. So the vectors normal to the faces are permutations of, from which the dihedral angles are calculated.

Simplices with an "orthogonal corner"

An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the Pythagorean theorem:
The sum of the squared -dimensional volumes of the facets adjacent to the orthogonal corner equals the squared -dimensional volume of the facet opposite of the orthogonal corner.
where are facets being pairwise orthogonal to each other but not orthogonal to, which is the facet opposite the orthogonal corner.
For a 2-simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron
with an orthogonal corner.

Relation to the (''n'' + 1)-hypercube

The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the -hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice. This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.
The n-simplex is also the vertex figure of the -hypercube. It is also the facet of the -orthoplex.

Topology

, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.

Probability

In probability theory, the points of the standard n-simplex in -space are the space of possible parameters of the categorical distribution on n + 1 possible outcomes.

Compounds

Since all simplices are self-dual, they can form a series of compounds;
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.
A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.
Note that each facet of an n-simplex is an affine -simplex, and thus the boundary of an n-simplex is an affine n − 1-chain. Thus, if we denote one positively oriented affine simplex as
with the denoting the vertices, then the boundary of σ is the chain
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
Likewise, the boundary of the boundary of a chain is zero: .
More generally, a simplex can be embedded into a manifold by means of smooth, differentiable map. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
where the are the integers denoting orientation and multiplicity. For the boundary operator, one has:
where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation.
A continuous map to a topological space X is frequently referred to as a singular n-simplex.

Algebraic geometry

Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine -dimensional space, where all coordinates sum up to 1. The algebraic description of this set is
which equals the scheme-theoretic description with
the ring of regular functions on the algebraic n-simplex.
By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one simplicial object, while the rings assemble into one cosimplicial object .
The algebraic n-simplices are used in higher K-theory and in the definition of higher Chow groups.

Applications