# simplex

simple

## WordNet

1. having only one part or element; "a simplex word has no affixes and is not part of a compound--like boy compared with boyish or house compared with houseboat"
2. allowing communication in only one direction at a time, or in telegraphy allowing only one message over a line at a time; "simplex system"
3. unornamented; "a simple country schoolhouse"; "her black dress--simple to austerity"
4. any herbaceous plant having medicinal properties
5. (botany) of leaf shapes; of leaves having no divisions or subdivisions (同)unsubdivided
6. having few parts; not complex or complicated or involved; "a simple problem"; "simple mechanisms"; "a simple design"; "a simple substance"

## PrepTutorEJDIC

1. 『簡単な』容易な,分かりやすい / (複合に対して)単一の / 『単純な』,込み入っていない / 『純然たる』,全くの / 『飾り気のない』,簡素な,地味な,質素な / 『もったいぶらない』;誠実な,実直な / お人よしの,だまされやすい / 《文》地位のない,普通の,平(ひら)の
2. 薬草,薬用植物

## Wikipedia preview

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By the second property the dot product of v0 with all other vectors is -13, 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 (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 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 (v0, ..., vn) is

where each column of the n × n determinant is the difference between the vectors representing two vertices.[6] 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

(so there are n! n-paths and does not depend on the permutation). 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.[7] In particular, the volume of such a simplex is

.

If P is a general parallelotope, the same assertions hold except that it is no more 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 a 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 obtains by observing that

From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) 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    (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),[8][9] while its central angle is cos−1(-1/n).[10]

### 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

Main article: Categorical distribution

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 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 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

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.[11]

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.[12]

• Complete graph
• Causal dynamical triangulation
• Distance geometry
• Delaunay triangulation
• Hill tetrahedron
• Other regular n-polytopes
• Hypercube
• Cross-polytope
• Tesseract
• 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

1. ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen Chapter IV, five dimensional semiregular polytope
2. ^ "Sloane's A135278 : Pascal's triangle with its left-hand edge removed". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)
4. ^ Yunmei Chen, Xiaojing Ye. "Projection Onto A Simplex". arXiv:1101.6081.
5. ^ MacUlan, N.; De Paula, G. G. (1989). "A linear-time median-finding algorithm for projecting a vector on the simplex of n". Operations Research Letters. 8 (4): 219. doi:10.1016/0167-6377(89)90064-3.
6. ^ A derivation of a very similar formula can be found in Stein, P. (1966). "A Note on the Volume of a Simplex". The American Mathematical Monthly. 73 (3): 299–301. doi:10.2307/2315353. JSTOR 2315353.
7. ^ Every n-path corresponding to a permutation is the image of the n-path by the affine isometry that sends to , and whose linear part matches to for all i. hence every two n-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the n-path is the set of points , with and Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit n-hypercube follows as well, replacing the strict inequalities above by "". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.
8. ^ Parks, Harold R.; Dean C. Wills (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex". The American Mathematical Monthly. Mathematical Association of America. 109 (8): 756–758. doi:10.2307/3072403.
9. ^ Harold R. Parks; Dean C. Wills (June 2009). Connections between combinatorics of permutations and algorithms and geometry. Oregon State University.
10. ^ Salvia, Raffaele (2013), Basic geometric proof of the relation between dimensionality of a regular simplex and its dihedral angle, arXiv:1304.0967
11. ^ Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. ISBN 0-471-07916-2.
12. ^ Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL-98-95: 1–32.

## 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.