Simplex
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In geometry, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. 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 . For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. 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 n-simplex 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.
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 (of the n+1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-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 (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 cross-polytope 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 1-faces (edges) of the n-simplex is the (n-1)th triangle number, the number of 2-faces of the n-simplex is the (n-2)th tetrahedron number, the number of 3-faces of the n-simplex is the (n-3)th 5-cell number, and so on.
Δ^{n} | Name | Schläfli symbol Coxeter-Dynkin |
0- faces (vertices) |
1- faces (edges) |
2- faces |
3- faces |
4- faces |
5- faces |
6- faces |
7- faces |
8- faces |
9- faces |
10- faces |
Sum =2^{n+1}-1 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Δ^{0} | 0-simplex (point) |
1 | 1 | |||||||||||
Δ^{1} | 1-simplex (line segment) |
{} Template:CDD |
2 | 1 | 3 | |||||||||
Δ^{2} | 2-simplex (triangle) |
{3} Template:CDD |
3 | 3 | 1 | 7 | ||||||||
Δ^{3} | 3-simplex (tetrahedron) |
{3,3} Template:CDD |
4 | 6 | 4 | 1 | 15 | |||||||
Δ^{4} | 4-simplex (5-cell) |
{3,3,3} Template:CDD |
5 | 10 | 10 | 5 | 1 | 31 | ||||||
Δ^{5} | 5-simplex | {3,3,3,3} Template:CDD |
6 | 15 | 20 | 15 | 6 | 1 | 63 | |||||
Δ^{6} | 6-simplex | {3,3,3,3,3} Template:CDD |
7 | 21 | 35 | 35 | 21 | 7 | 1 | 127 | ||||
Δ^{7} | 7-simplex | {3,3,3,3,3,3} Template:CDD |
8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | 255 | |||
Δ^{8} | 8-simplex | {3,3,3,3,3,3,3} Template:CDD |
9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | 511 | ||
Δ^{9} | 9-simplex | {3,3,3,3,3,3,3,3} Template:CDD |
10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | 1023 | |
Δ^{10} | 10-simplex | {3,3,3,3,3,3,3,3,3} Template:CDD |
11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | 2047 |
In some conventions,^{[3]} 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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The standard simplex
The standard n-simplex (or unit n-simplex) 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 n+1 vertices of the standard n-simplex are the points e_{i} ∈ R^{n+1}, where
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v_{0}, …, v_{n}) given by
The coefficients t_{i} 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 (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 n-tuples between 0 and 1:
Geometrically, this is an n-dimensional subset of (maximal dimension, codimension 0) rather than of (codimension 1). 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 (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 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 (disjoint except for boundaries), 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
can be easily calculated from sorting .^{[4]} The sorting approach takes complexity, which can be improved to complexity via median-finding algorithms.^{[5]} 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,
- For a regular simplex, the distances of its vertices to its center are equal.
- The angle subtended by any two vertices of an n-dimensional simplex through its center is
These can be used as follows. Let vectors (v_{0}, v_{1}, ..., v_{n}) 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 (v_{0}, v_{1}, v_{2}, v_{3}) are the vertices of a 3-simplex 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 -Template:Frac, 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
Volume
The oriented volume of an n-simplex in n-dimensional 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. A derivation of a very similar formula can be found in.^{[6]} Without the 1/n! it is the formula for the volume of an n-parallelepiped. One way to understand the 1/n! factor is as follows. If the coordinates of a point in a unit n-box 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 (volume-preserving) transformation, but sorting compressed the space by a factor of n!.
The volume under a standard n-simplex (i.e. between the origin and the simplex in R^{n+1}) is
The volume of a regular n-simplex with unit side length is
as can be seen by multiplying the previous formula by x^{n+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 (where the n-simplex side length is 1), and normalizing by the length of the increment, , along the normal vector.
The dihedral angle of a regular n-dimensional simplex is cos^{−1}(1/n),^{[7]}^{[8]} while its central angle is cos^{−1}(-1/n).^{[9]}
Simplexes with an "orthogonal corner"
Orthogonal corner means here, that there is a vertex at which all adjacent facets are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists an n-dimensional version of the Pythagorean theorem:
The sum of the squared (n-1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n-1)-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 a cube 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 (n+1)-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 (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 n-simplex is also the vertex figure of the (n+1)-hypercube. It is also the facet of the (n+1)-orthoplex.
Topology
Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.
Probability
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In probability theory, the points of the standard n-simplex in -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 k-simplexes embedded in an open subset of R^{n} 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 n-1-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 (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 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 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 n+1-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
which equals the scheme-theoretic description with
the ring of regular functions on the algebraic n-simplex (for any ring ).
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 (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).
The algebraic n-simplices are used in higher K-Theory and in the definition of higher Chow groups.
Applications
Template:Expand section 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.^{[10]}
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.^{[11]}
See also
- Causal dynamical triangulation
- Distance geometry
- Delaunay triangulation
- Hill tetrahedron
- Other regular n-polytopes
- Hypersimplex
- 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
- 3-sphere
Notes
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }} Chapter IV, five dimensional semiregular polytope
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- ↑ Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)
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References
- Walter Rudin, Principles of Mathematical Analysis (Third Edition), (1976) McGraw-Hill, New York, ISBN 0-07-054235-X (See chapter 10 for a simple review of topological properties.).
- Andrew S. Tanenbaum, Computer Networks (4th Ed), (2003) Prentice Hall, ISBN 0-13-066102-3 (See 2.5.3).
- Luc Devroye, Non-Uniform Random Variate Generation. (1986) ISBN 0-387-96305-7; Web version freely downloadable.
- H.S.M. Coxeter, Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8
- p120-121
- p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
- Weisstein, Eric W., "Simplex", MathWorld.
- Stephen Boyd and Lieven Vandenberghe, Convex Optimization, (2004) Cambridge University Press, New York, NY, USA.