 Simplex

For other uses, see Simplex (disambiguation).
In geometry, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an nsimplex is an ndimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is a pentachoron. A single point may be considered a 0simplex, and a line segment may be considered a 1simplex. A simplex may be defined as the smallest convex set containing the given vertices.
A regular simplex^{[1]} is a simplex that is also a regular polytope. A regular nsimplex may be constructed from a regular (n − 1)simplex by connecting a new vertex to all original vertices by the common edge length.
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.
Contents
Elements
The convex hull of any nonempty subset of the n+1 points that define an nsimplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m+1 (of the n+1 defining points) is an msimplex, called an mface of the nsimplex. The 0faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1faces are called the edges, the (n − 1)faces are called the facets, and the sole nface is the whole nsimplex itself. In general, the number of mfaces is equal to the binomial coefficient . Consequently, the number of mfaces of an nsimplex may be found in column (m + 1) of row (n + 1) 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 Simplicial complex#Definitions
The regular simplex family is the first of three regular polytope families, labeled by Coxeter as α_{n}, the other two being the crosspolytope family, labeled as β_{n}, and the hypercubes, labeled as γ_{n}. A fourth family, the infinite tessellation of hypercubes, he labeled as δ_{n}.
The number of 1faces (edges) of the nsimplex is the (n1)th triangle number, the number of 2faces (faces) of the nsimplex is the (n2)th tetrahedron number, the number of 3faces (cells) of the nsimplex is the (n3)th pentachoron number, and so on.
nSimplex elements^{[2]} Δ^{n} Name Schläfli symbol
CoxeterDynkin0
faces
(vertices)1
faces
(edges)2
faces
(faces)3
faces
(cells)4
faces
5
faces
6
faces
7
faces
8
faces
9
faces
10
faces
Sum
=2^{n+1}1Δ^{0} 0simplex
(point)1 1 Δ^{1} 1simplex
(line segment){}
2 1 3 Δ^{2} 2simplex
(triangle){3}
3 3 1 7 Δ^{3} 3simplex
(tetrahedron){3,3}
4 6 4 1 15 Δ^{4} 4simplex
(pentachoron){3,3,3}
5 10 10 5 1 31 Δ^{5} 5simplex
(hexateron){3,3,3,3}
6 15 20 15 6 1 63 Δ^{6} 6simplex
(heptapeton){3,3,3,3,3}
7 21 35 35 21 7 1 127 Δ^{7} 7simplex
(octaexon){3,3,3,3,3,3}
8 28 56 70 56 28 8 1 255 Δ^{8} 8simplex
(enneazetton){3,3,3,3,3,3,3}
9 36 84 126 126 84 36 9 1 511 Δ^{9} 9simplex
(decayotton){3,3,3,3,3,3,3,3}
10 45 120 210 252 210 120 45 10 1 1023 Δ^{10} 10simplex
(hendecaxennon){3,3,3,3,3,3,3,3,3}
11 55 165 330 462 462 330 165 55 11 1 2047 In some conventions,^{[who?]} the empty set is defined to be a (−1)simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
Symmetric graphs of regular simplices
These Petrie polygon (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.
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20The standard simplex
The standard nsimplex (or unit nsimplex) is the subset of R^{n+1} given by
The simplex Δ^{n} lies in the affine hyperplane obtained by removing the restriction t_{i} ≥ 0 in the above definition. The standard simplex is clearly regular.
The n+1 vertices of the standard nsimplex are the points {e_{i}} ⊂ R^{n+1}, where
 e_{0} = (1, 0, 0, ..., 0),
 e_{1} = (0, 1, 0, ..., 0),
 e_{n} = (0, 0, 0, ..., 1).
There is a canonical map from the standard nsimplex to an arbitrary nsimplex with vertices (v_{0}, …, v_{n}) given by
The coefficients t_{i} are called the barycentric coordinates of a point in the nsimplex. Such a general simplex is often called an affine nsimplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine nsimplex to emphasize that the canonical map may be orientation preserving or reversing.
More generally, there is a canonical map from the standard (n − 1)simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex:
Increasing coordinates
An alternative coordinate system is given by taking the indefinite sum:
This yields the alternative presentation by order, namely as nondecreasing ntuples between 0 and 1:
Geometrically, this is an ndimensional subset of (maximal dimension, codimension 0) rather than of (codimension 1). The hyperfaces, which on the standard simplex correspond to one coordinate vanishing, t_{i} = 0, here correspond to successive coordinates being equal, s_{i} = s_{i + 1}, while the interior corresponds to the inequalities becoming strict (increasing sequences).
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 (closed) fundamental domain for the action of the symmetric group on the ncube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the ncube into n! mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume 1 / n! 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 infinitedimensional 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 Δ is chosen such that
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 nsimplex as a corner of the ncube, 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 faces.
Cartesian coordinates for regular ndimensional simplex in R^{n}
The coordinates of the vertices of a regular ndimensional simplex can be obtained from these two properties,
 For a regular simplex, the distances of its vertices to its center are equal.
 The angle subtended by any two vertices of an ndimensional simplex through its center is
These can be used as follows. Let vectors (v_{0}, v_{1}, ..., v_{n}) represent the vertices of an nsimplex 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 ^{1}⁄_{n}. This can be used to calculate positions for them.
For example in three dimensions the vectors (v_{0}, v_{1}, v_{2}, v_{3}) are the vertices of a 3simplex or tetrahedron. Write these as
Choose the first vector v_{0} to have all but the first component zero, so by the first property it must be (1, 0, 0) and the vectors become
By the second property the dot product of v_{0} with all other vectors is ^{1}⁄_{3}, so each of their x components must equal this, and the vectors become
Next choose v_{1} 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 (choose any of the two square roots), 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 v_{1} 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
The oriented volume of an nsimplex in ndimensional space with vertices (v_{0}, ..., v_{n}) is
where each column of the n × n determinant is the difference between the vectors representing two vertices. Without the 1/n! it is the formula for the volume of an nparallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit nbox are sorted, together with 0 and 1, and successive differences are taken, then since the results add to one, the result is a point in an n simplex spanned by the origin and the closest n vertices of the box. The taking of differences was a unimodular (volumepreserving) transformation, but sorting compressed the space by a factor of n!.
The volume under a standard nsimplex (i.e. between the origin and the simplex in R^{n+1}) is
The volume of a regular nsimplex with unit side length is
as can be seen by multiplying the previous formula by x^{n+1}, to get the volume under the nsimplex as a function of its vertex distance x from the origin, differentiating with respect to x, at (where the nsimplex side length is 1), and normalizing by the length of the increment, , along the normal vector.
The dihedral angle of a regular ndimensional simplex is cos^{−1}(1/n).^{[3]}^{[4]}
Simplexes with an "orthogonal corner"
Orthogonal corner means here, that there is a vertex at which all adjacent hyperfaces are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists a ndimensional version of the Pythagorean theorem:
The sum of the squared (n1)dimensional volumes of the hyperfaces adjacent to the orthogonal corner equals the squared (n1)dimensional volume of the hyperface opposite of the orthogonal corner.
where are hyperfaces being pairwise orthogonal to each other but not orthogonal to A_{0}, which is the hyperface opposite of the orthogonal corner.
For a 2simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3simplex it is de Gua's theorem for a tetrahedron with a cube corner.
Relation to the (n+1)hypercube
The Hasse diagram of the face lattice of an nsimplex is isomorphic to the graph of the (n+1)hypercube's edges, with the hypercube's vertices mapping to each of the nsimplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). 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 nsimplex is also the vertex figure of the (n+1)hypercube. It is also the facet of the (n+1)orthoplex.
Topology
Topologically, an nsimplex is equivalent to an nball. Every nsimplex is an ndimensional manifold with boundary.
Probability
Main article: Categorical distributionIn probability theory, the points of the standard nsimplex in (n + 1)space are the space of possible parameters (probabilities) of the categorical distribution on n+1 possible outcomes.
Algebraic topology
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 ksimplexes embedded in an open subset of R^{n} is called an affine kchain. 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 face of an nsimplex is an affine n1simplex, and thus the boundary of an nsimplex is an affine n1chain. Thus, if we denote one positivelyoriented affine simplex as
 σ = [v_{0},v_{1},v_{2},...,v_{n}]
with the v_{j} denoting the vertices, then the boundary of σ is the chain
 .
More generally, a simplex (and a chain) 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 a_{i} 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 (by definition of a map).
A continuous map to a topological space X is frequently referred to as a singular nsimplex.
Applications
Simplices are used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot.
In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming.^{[5]}
In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.
In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.^{[6]}
See also
 Causal dynamical triangulation
 Distance geometry
 Delaunay triangulation
 Hill tetrahedron
 Other regular npolytopes
 Polytope
 Metcalfe's Law
 List of regular polytopes
 Schläfli orthoscheme
 Simplex algorithm  a method for solving optimisation problems with inequalities.
 Simplicial complex
 Simplicial homology
 Simplicial set
 Ternary plot
 3sphere
Notes
 ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen Chapter IV, five dimensional semiregular polytope
 ^ (sequence A135278 in OEIS)
 ^ Harold R. Parks; Dean C. Wills (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular nSimplex". The American Mathematical Monthly (Mathematical Association of America) 109 (8): 756–758. http://www.jstor.org/stable/3072403.
 ^ Harold R. Parks; Dean C. Wills (June 2009). Connections between combinatorics of permutations and algorithms and geometry. Oregon State University. http://ir.library.oregonstate.edu/xmlui/handle/1957/11929.
 ^ Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. ISBN 0471079162.
 ^ Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report HPL9895: 1–32. http://www.hpl.hp.com/techreports/98/HPL9895.pdf.
References
 Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGrawHill, New York, ISBN 007054235X (See chapter 10 for a simple review of topological properties.).
 Andrew S. Tanenbaum, Computer Networks (4th Ed), (2003) Prentice Hall, ISBN 0130661023 (See 2.5.3).
 Luc Devroye, NonUniform Random Variate Generation. (1986) ISBN 0387963057; Web version freely downloadable.
 H.S.M. Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0486614808
 p120121
 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n>=5)
 Weisstein, Eric W., "Simplex" from MathWorld.
External links
 Olshevsky, George, Simplex at Glossary for Hyperspace.
 OEIS sequence A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1,m+1); or, Pascal's triangle A007318 with its lefthand edge removed.
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