simplex

• adj.

• 単一性の、単式

関

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

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/10/27 20:57:42」(JST)

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For other uses, see Simplex (disambiguation).

A regular 3-simplex or tetrahedron

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 $u_{0},\dots ,u_{k}\in \mathbb {R} ^{k}$ are affinely independent, which means $u_{1}-u_{0},\dots ,u_{k}-u_{0}$ are linearly independent. Then, the simplex determined by them is the set of points

$C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\bigg |}~\theta _{i}\geq 0,0\leq i\leq k,\sum _{i=0}^{k}\theta _{i}=1\right\}$.

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 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.

Examples

• A 0-simplex is a point.
• A 1-simplex is a line segment.
• A 2-simplex is a triangle.
• A 3-simplex is a tetrahedron.

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 ${\tbinom {n+1}{m+1}}$. 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 simplical complex for more detail.

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-th triangle number, the number of 2-faces of the n-simplex is the (n-1)th tetrahedron number, the number of 3-faces of the n-simplex is the (n-2)th 5-cell number, and so on.

n-Simplex elements

Δn	Name	Schläfli Coxeter	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 =2n+1-1
Δ0	0-simplex (point)	( )	1											1
Δ1	1-simplex (line segment)	{ } = ( )∨( ) = 2.( )	2	1										3
Δ2	2-simplex (triangle)	{3} = 3.( )	3	3	1									7
Δ3	3-simplex (tetrahedron)	{3,3} = 4.( )	4	6	4	1								15
Δ4	4-simplex (5-cell)	{33} = 5.( )	5	10	10	5	1							31
Δ5	5-simplex	{34} = 6.( )	6	15	20	15	6	1						63
Δ6	6-simplex	{35} = 7.( )	7	21	35	35	21	7	1					127
Δ7	7-simplex	{36} = 8.( )	8	28	56	70	56	28	8	1				255
Δ8	8-simplex	{37} = 9.( )	9	36	84	126	126	84	36	9	1			511
Δ9	9-simplex	{38} = 10.( )	10	45	120	210	252	210	120	45	10	1		1023
Δ10	10-simplex	{39} = 11.( )	11	55	165	330	462	462	330	165	55	11	1	2047

An (n+1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ). An (m+n+1)-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 a joint of two points: ( )∨( ) = 2.( ). A general 2-simplex (scalene triangle) is the join of 3 points: ( )∨( )∨( ). An isosceles triangle is the join of a 1-simplex and a point: { }∨( ). An equilateral triangle is 3.( ) or {3}. 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 2 points: { }∨( )∨( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4.( ) or {3,3} and so on.

The total number of faces is always a power of two minus one. This figure (a projection of the tesseract) shows the centroids of the 15 faces of the tetrahedron.

The numbers of faces in the above table are the same as in Pascal's triangle, without the left diagonal.

In some conventions, 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 polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20

The standard simplex

The standard 2-simplex in R³

The standard n-simplex (or unit n-simplex) is the subset of Rⁿ⁺¹ given by

$\Delta ^{n}=\left\{(t_{0},\cdots ,t_{n})\in \mathbb {R} ^{n+1}\mid \sum _{i=0}^{n}{t_{i}}=1{\mbox{ and }}t_{i}\geq 0{\mbox{ for all }}i\right\}$

The simplex Δⁿ 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 ∈ Rⁿ⁺¹, where

e₀ = (1, 0, 0, ..., 0),

e₁ = (0, 1, 0, ..., 0),

$\vdots$

e_n = (0, 0, 0, ..., 1).

There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v₀, …, v_n) given by

$(t_{0},\cdots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}$

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 $(n-1)$-simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):

$(t_{1},\cdots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}$

These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: $\Delta ^{n-1}\twoheadrightarrow P.$

Examples

• Δ⁰ is the point 1 in R¹.
• Δ¹ is the line segment joining (1,0) and (0,1) in R².
• Δ² is the equilateral triangle with vertices (1,0,0), (0,1,0) and (0,0,1) in R³.
• Δ³ is the regular tetrahedron with vertices (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) in R⁴.

Increasing coordinates

An alternative coordinate system is given by taking the indefinite sum:

${\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\dots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\dots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\dots +t_{n}=1\end{aligned}}$

This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:

$\Delta _{*}^{n}=\left\{(s_{1},\cdots ,s_{n})\in \mathbb {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.$

Geometrically, this is an n-dimensional subset of $\mathbb {R} ^{n}$ (maximal dimension, codimension 0) rather than of $\mathbb {R} ^{n+1}$ (codimension 1). The facets, 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 n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube 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 $1,x,x^{2}/2,x^{3}/3!,\dots ,x^{n}/n!$

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 $\,(p_{i})_{i}$ with possibly negative entries, the closest point $\left(t_{i}\right)_{i}$ on the simplex has coordinates

$t_{i}=\max\{p_{i}+\Delta \,,0\},$

where $\Delta$ is chosen such that $\sum _{i}\max\{p_{i}+\Delta \,,0\}=1.$

$\Delta$ can be easily calculated from sorting $p_{i}$. The sorting approach takes $O(n\log n)$ complexity, which can be improved to $O(n)$ complexity via median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the $\ell _{1}$ 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:

$\Delta _{c}^{n}=\left\{(t_{1},\cdots ,t_{n})\in \mathbb {R} ^{n}\mid \sum _{i=1}^{n}{t_{i}}\leq 1{\mbox{ and }}t_{i}\geq 0{\mbox{ for all }}i\right\}.$

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 Rⁿ

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 $\arccos \left({\tfrac {-1}{n}}\right)$

These can be used as follows. Let vectors (v₀, v₁, ..., 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 $-1/n$. This can be used to calculate positions for them.

For example in three dimensions the vectors (v₀, v₁, v₂, v₃) are the vertices of a 3-simplex or tetrahedron. Write these as

${\begin{pmatrix}x_{0}\\y_{0}\\z_{0}\end{pmatrix}},{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}x_{3}\\y_{3}\\z_{3}\end{pmatrix}}$

Choose the first vector v₀ to have all but the first component zero, so by the first property it must be (1, 0, 0) and the vectors become

${\begin{pmatrix}1\\\end{pmatrix}},{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}x_{3}\\y_{3}\\z_{3}\end{pmatrix}}\\end{pmatrix}},{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}x_{3}\\y_{3}\\z_{3}\end{pmatrix}}}$

By the second property the dot product of v₀ with all other vectors is - ¹⁄₃, so each of their x components must equal this, and the vectors become

${\begin{pmatrix}1\\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}}$

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:

${\begin{pmatrix}1\\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}}$

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

${\begin{pmatrix}1\\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{3}\end{pmatrix}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{3}\end{pmatrix}}}$

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

${\begin{pmatrix}1\\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\{\sqrt {\frac {2}{3}}}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\-{\sqrt {\frac {2}{3}}}\end{pmatrix}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\{\sqrt {\frac {2}{3}}}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\-{\sqrt {\frac {2}{3}}}\end{pmatrix}}}$

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, ..., v_n) is

$\left|{1 \over n!}\det {\begin{pmatrix}v_{1}-v_{0},&v_{2}-v_{0},&\dots ,&v_{n}-v_{0}\end{pmatrix}}\right|$

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 n-parallelotope. This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis $(v_{0},e_{1},\ldots ,e_{n})$ of $\mathbf {R} ^{n}$. Given a permutation $\sigma$ of $\{1,2,\ldots ,n\}$, call a list of vertices $v_{0},\ v_{1},\ldots ,v_{n}$ a n-path if

$v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}$

(so there are n! n-paths and $v_{n}$ 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. In particular, the volume of such a simplex is

${\rm {Vol}}\ (P)/n!=1/n!$.

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 $\mathbf {R} ^{n}$ to $e_{1},\ldots ,e_{n}$. As previously, this implies that the volume of a simplex coming from a n-path is:

${\rm {Vol}}(P)/n!=\det(e_{1},\ldots ,e_{n})/n!.$

Conversely, given a n-simplex $(v_{0},\ v_{1},\ v_{2},\ldots v_{n})$ of $\mathbf {R} ^{n}$, it can be supposed that the vectors $e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}$ form a basis of $\mathbf {R} ^{n}$. Considering the parallelotope constructed from $v_{0}$ and $e_{1},\ldots ,e_{n}$, one sees that the previous formula is valid for every simplex.

Finally, the formula at the beginning of this section obtains by observing that

$\det(v_{1}-v_{0},v_{2}-v_{0},\ldots v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).$

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

${1 \over (n+1)!}$

The volume of a regular n-simplex with unit side length is

${\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}$

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 $x=1/{\sqrt {2}}$   (where the n-simplex side length is 1), and normalizing by the length $dx/{\sqrt {n+1}}\,$ of the increment, $(dx/(n+1),\dots ,dx/(n+1))$, along the normal vector.

The dihedral angle of a regular n-dimensional simplex is cos⁻¹(1/n), while its central angle is cos⁻¹(-1/n).

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.

$\sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}$

where $A_{1}\ldots A_{n}$ are facets being pairwise orthogonal to each other but not orthogonal to $A_{0}$, 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 $(n+1)$-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 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

$\sigma =[v_{0},v_{1},v_{2},...,v_{n}]$

with the $v_{j}$ denoting the vertices, then the boundary $\partial \sigma$ of σ is the chain

$\partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1},...,v_{n}]$.

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:

$\partial ^{2}\sigma =\partial (~\sum _{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1},...,v_{n}]~)=0.$

Likewise, the boundary of the boundary of a chain is zero: $\partial ^{2}\rho =0$.

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map $f\colon \mathbb {R} ^{n}\rightarrow M$. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

$f(\sum \nolimits _{i}a_{i}\sigma _{i})=\sum \nolimits _{i}a_{i}f(\sigma _{i})$

where the $a_{i}$ are the integers denoting orientation and multiplicity. For the boundary operator $\partial$, one has:

$\partial f(\rho )=f(\partial \rho )$

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 $f:\sigma \rightarrow X$ 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

$\Delta ^{n}:=\{x\in \mathbb {A} ^{n+1}\vert \sum _{i=1}^{n+1}x_{i}-1=0\}$,

which equals the scheme-theoretic description $\Delta _{n}(R)=Spec(R[\Delta ^{n}])$ with

$R[\Delta ^{n}]:=R[x_{1},...,x_{n+1}]/(\sum x_{i}-1)$

the ring of regular functions on the algebraic n-simplex (for any ring $R$).

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 $R[\Delta ^{n}]$ assemble into one cosimplicial object $R[\Delta ^{\bullet }]$ (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.

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.

See also

• 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

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.

External links

• Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007. 

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}x_{3}\\y_{3}\\z_{3}\end{pmatrix}}}</annotation>

 </semantics>

</math>

By the second property the dot product of v₀ with all other vectors is - ¹⁄₃, so each of their x components must equal this, and the vectors become

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\<add_contents_exp><m=6 date=20150923>\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}}</annotation> </semantics> </math>

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:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>8</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\<add_contents_exp><m=6 date=20150923>\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{2}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\y_{3}\\z_{3}\end{pmatrix}}}</annotation> </semantics> </math>

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

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>8</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\<add_contents_exp><m=6 date=20150923>\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{2}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\z_{3}\end{pmatrix}}}</annotation> </semantics> </math>

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

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>8</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1\<add_contents_exp><m=6 date=20150923>\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\{\frac {\sqrt {8}}{3}}\<add_contents_exp><m=6 date=20150923>\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\{\sqrt {\frac {2}{3}}}\end{pmatrix}},{\begin{pmatrix}-{\frac {1}{3}}\\-{\frac {\sqrt {2}}{3}}\\-{\sqrt {\frac {2}{3}}}\end{pmatrix}}}</annotation> </semantics> </math>

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, ..., v_n) is

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mo>…</mo> <mo>,</mo> </mtd> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{1 \over n!}\det {\begin{pmatrix}v_{1}-v_{0},&v_{2}-v_{0},&\dots ,&v_{n}-v_{0}\end{pmatrix}}\right|}</annotation> </semantics> </math>

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 n-parallelotope. This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{0},e_{1},\ldots ,e_{n})}</annotation> </semantics> </math> of <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math>. Given a permutation <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math> of <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation> </semantics> </math>, call a list of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0},\ v_{1},\ldots ,v_{n}}</annotation> </semantics> </math> a n-path if

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}}</annotation> </semantics> </math>

(so there are n! n-paths and <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{n}}</annotation> </semantics> </math> 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. In particular, the volume of such a simplex is

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {Vol}}\ (P)/n!=1/n!}</annotation> </semantics> </math>.

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 <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math> to <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation> </semantics> </math>. As previously, this implies that the volume of a simplex coming from a n-path is:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {Vol}}(P)/n!=\det(e_{1},\ldots ,e_{n})/n!.}</annotation> </semantics> </math>

Conversely, given a n-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (v_{0},\ v_{1},\ v_{2},\ldots v_{n})}</annotation> </semantics> </math> of <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math>, it can be supposed that the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}}</annotation> </semantics> </math> form a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n}}</annotation> </semantics> </math>. Considering the parallelotope constructed from <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{0}}</annotation> </semantics> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation> </semantics> </math>, one sees that the previous formula is valid for every simplex.

Finally, the formula at the beginning of this section obtains by observing that

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(v_{1}-v_{0},v_{2}-v_{0},\ldots v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).}</annotation> </semantics> </math>

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

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1 \over (n+1)!}}</annotation> </semantics> </math>

The volume of a regular n-simplex with unit side length is

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> <mrow> <mi>n</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}}</annotation> </semantics> </math>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1/{\sqrt {2}}}</annotation> </semantics> </math>   (where the n-simplex side length is 1), and normalizing by the length <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx/{\sqrt {n+1}}\,}</annotation> </semantics> </math> of the increment, <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (dx/(n+1),\dots ,dx/(n+1))}</annotation> </semantics> </math>, along the normal vector.

The dihedral angle of a regular n-dimensional simplex is cos⁻¹(1/n), while its central angle is cos⁻¹(-1/n).

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.

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}}</annotation> </semantics> </math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}\ldots A_{n}}</annotation> </semantics> </math> are facets being pairwise orthogonal to each other but not orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math>, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+1)}</annotation> </semantics> </math>-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

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =[v_{0},v_{1},v_{2},...,v_{n}]}</annotation> </semantics> </math>

with the <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{j}}</annotation> </semantics> </math> denoting the vertices, then the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \sigma }</annotation> </semantics> </math> of σ is the chain

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>σ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1},...,v_{n}]}</annotation> </semantics> </math>.

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>σ</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">[</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mtext> </mtext> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{2}\sigma =\partial (~\sum _{j=0}^{n}(-1)^{j}[v_{0},...,v_{j-1},v_{j+1},...,v_{n}]~)=0.}</annotation> </semantics> </math>

Likewise, the boundary of the boundary of a chain is zero: <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{2}\rho =0}</annotation> </semantics> </math>.

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} ^{n}\rightarrow M}</annotation> </semantics> </math>. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\sum \nolimits _{i}a_{i}\sigma _{i})=\sum \nolimits _{i}a_{i}f(\sigma _{i})}</annotation> </semantics> </math>

where the <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math> are the integers denoting orientation and multiplicity. For the boundary operator <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial }</annotation> </semantics> </math>, one has:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial f(\rho )=f(\partial \rho )}</annotation> </semantics> </math>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>σ</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\sigma \rightarrow X}</annotation> </semantics> </math> 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

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">|</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ^{n}:=\{x\in \mathbb {A} ^{n+1}\vert \sum _{i=1}^{n+1}x_{i}-1=0\}}</annotation> </semantics> </math>,

which equals the scheme-theoretic description <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mi>p</mi> <mi>e</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{n}(R)=Spec(R[\Delta ^{n}])}</annotation> </semantics> </math> with

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>:=</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mo>∑</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{n}]:=R[x_{1},...,x_{n+1}]/(\sum x_{i}-1)}</annotation> </semantics> </math>

the ring of regular functions on the algebraic n-simplex (for any ring <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math>).

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 <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{n}]}</annotation> </semantics> </math> assemble into one cosimplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\Delta ^{\bullet }]}</annotation> </semantics> </math> (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.

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.

See also

• 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

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.

External links

• Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007. 

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

</raw> </toggledisplay>

English Journal

• Size-exclusion chromatography (HPLC-SEC) technique optimization by simplex method to estimate molecular weight distribution of agave fructans.

• Moreno-Vilet L1, Bostyn S2, Flores-Montaño JL3, Camacho-Ruiz RM4.
• Food chemistry.Food Chem.2017 Dec 15;237:833-840. doi: 10.1016/j.foodchem.2017.06.020. Epub 2017 Jun 7.
• PMID 28764075

• Loratadine bioavailability via buccal transferosomal gel: formulation, statistical optimization, in vitro/in vivo characterization, and pharmacokinetics in human volunteers.

• Elkomy MH1, El Menshawe SF1, Abou-Taleb HA2, Elkarmalawy MH2.
• Drug delivery.Drug Deliv.2017 Nov;24(1):781-791. doi: 10.1080/10717544.2017.1321061.
• PMID 28480758

• Seek and destroy: targeted adeno-associated viruses for gene delivery to hepatocellular carcinoma.

• Dhungel B1,2,3, Jayachandran A1,2, Layton CJ2,4, Steel JC1,2.
• Drug delivery.Drug Deliv.2017 Nov;24(1):289-299. doi: 10.1080/10717544.2016.1247926.
• PMID 28165834

• Low-molecular weight mannogalactofucans prevent herpes simplex virus type 1 infection via activation of Toll-like receptor 2.

• Kim WJ1, Choi JW2, Jang WJ3, Kang YS4, Lee CW2, Synytsya A5, Park YI6.
• International journal of biological macromolecules.Int J Biol Macromol.2017 Oct;103:286-293. doi: 10.1016/j.ijbiomac.2017.05.060. Epub 2017 May 15.
• PMID 28522392

Japanese Journal

• 症例報告 髄液オレキシン低値をともない過眠症を合併した単純ヘルペス脳炎の1例

• 向野 晃弘,木下 郁夫,福島 直美 [他]
• 臨床神経学 = Clinical neurology 54(3), 207-211, 2014-03
• NAID 40020006443

• Axin expression delays herpes simplex virus-induced autophagy and enhances viral replication in L929 cells

• Choi Eun-Jin,Kee Sun-Ho
• Microbiology and immunology 58(2), 103-111, 2014-02
• NAID 40020007912

• Eight types of stem cells in the life cycle of the moss Physcomitrella patens

• Kofuji Rumiko,Hasebe Mitsuyasu
• Current Opinion in Plant Biology 17(1), 13-21, 2014-02
• … A simplex meristem with a single stem cell was acquired in the sporophyte generation early in land plant evolution. … patens develops at least seven types of simplex meristem in the gametophyte and at least one type in the sporophyte generation and is a good material for regulatory network comparisons. …
• NAID 120005418471

• 前頭葉に病変が及んだ非ヘルペス性急性辺縁系脳炎の1例

• 朱膳寺 圭子,石川 元直,西村 芳子,柴田 興一,大塚 邦明,佐倉 宏,高橋 幸利
• 東京女子医科大学雑誌 84(E1), E197-E203, 2014-01-31
• 【症例】45歳男性【主訴】発熱、痙攣発作、意識障害【現病歴】入院2週間前から感冒症状が出現し、入院1週間前から40℃の発熱があり解熱鎮痛剤を内服していた。入院当日、全身性の痙攣発作が出現したため救急搬送された。【入院後経過】入院時、軽度の意識障害と前向性健忘を認め、頭部MRIでは異常はなかったが、第2病日、口からはじまる部分発作が増悪した。第6病日に強直性痙攣が出現し、第7病日のMRIでは両側の海 …
• NAID 110009752578

Related Links

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Merger through corporate reorganization of SCK Holdings Co., Ltd., Simplex Holdings, Inc., and Simplex Consulting, Inc.

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★リンクテーブル★

リンク元	「100Cases 33」「100Cases 16」「単一性」「単式」
拡張検索	「herpes simplex virus protein Vmw65」「herpes simplex virus vaccine」「herpes simplex genitalis」
関連記事	「simple」

「100Cases 33」

　　[★]

☆case33　頭痛と混乱

■glossary

accompany

vt.

(人)と同行する、(人)に随行する。(もの)に付随する。～と同時に起こる。～に加える(添える、同封する)(with)

slurred n. 不明瞭

強直間代痙攣 tonic-clonic convulsion

　意識消失とともに全身の随意筋に強直痙攣が生じ(強直痙攣期tonic convulsion)、次いで全身の筋の強直と弛緩とが律動的に繰り返される時期(間代痙攣期clonic convulsion)を経て、発作後もうろう状態を呈する一連の発作。

■症例

28歳、女性　黒人　南アフリカ　手術室看護師　ロンドン住在

主訴：頭痛と混乱

現病歴：過去3週間で頭痛が続いており、ひどくなってきた。現在も頭痛が持続しており、頭全体が痛い。友人曰く「過去六ヶ月で体重が10kg減っていて、最近、混乱してきたようだ」。発話は不明瞭。救急室にいる間に強直間代痙攣を起こした。

・診察 examination

やせている。55kg。38.5℃。口腔カンジダ症(oral candidiasis)。リンパ節腫脹無し。心血管、呼吸器系、消化器系正常。痙攣前における神経検査では時間、場所、人の見当識無し。神経局所症状無し(no focal neurological sign)。眼底両側に乳頭浮腫有り。

・検査 investigation

血算：白血球増多

血液生化学：ナトリウム低下

CT：供覧

■キーワード＆着目するポイント

・口腔カンジダ症(oral candidiasis)

・頭痛、精神症状、強直間代痙攣

・眼底両側に乳頭浮腫

・CT所見

・低ナトリウム血症は二次的なもの

■アプローチ

・口腔カンジダ症(oral candidiasis)　→　細胞免疫低下状態(DM、免疫抑制、AIDSなど) or 常在細菌叢の攪乱(長期の抗菌薬の使用)

　・The occurrence of thrush in a young, otherwise healthy-appearing person should prompt an investigation for underlying HIV infection.(HIM.1254)

　・More commonly, thrush is seen as a nonspecific manifestation of severe debilitating illness.(HIM.1254)

・精神症状、強直間代痙攣　→　一次的、あるいは二次的な脳の疾患がありそう

・頭痛　→　漠然としていて絞れないが、他の症状からして機能性頭痛ではなく症候性頭痛っぽい。

・眼底両側に乳頭浮腫　→　脳圧亢進の徴候　→　原因は・・・脳腫瘍、ことにテント下腫瘍と側頭葉の腫瘍、クモ膜下出血、脳水腫など、そのほか、眼窩内病変、低眼圧などの局所的要因、悪性高血圧、血液疾患、大量出血、肺気腫などの全身的要因 (vindicate本のp342も参考になる)

　・頭痛と脳圧亢進　→　頭蓋内圧占拠性病変、脳炎(IMD.274)

・CT所見　→　ringformの病変、脳浮腫、脳圧亢進

・低ナトリウム血症　→　脳ヘルニアに続発して起こることがあるらしい。実際には下垂体にトキソプラズマによる病変が形成されることにより起こりうる。

・そのほか出身地、体重減少もHIVを疑わせる点

パターン認識でHIV + 精神症状 + てんかん発作(強直間代痙攣) + 脳圧亢進 + CT所見 = 一番ありそうなのはToxoplasma gondiiによるトキソプラズマ脳症 cerebral toxoplasosis (トキソプラズマ脳炎 toxoplasmic encephalitis)

■Toxoplasma gondii

　原虫　胞子原虫類

(感染予防学 080521のプリント、CASES p,92、HIM p.1305-)

・疫学：西洋では30-80%の成人がトキソプラズマの感染の既往がある・・・うぇ(CASES)。日本では10%前後(Wikipedia)。

・生活環

　・終宿主：ネコ：ネコの小腸上皮細胞で有性・無性生殖　糞便にオーシストの排泄

　・中間宿主：ヒト.ブタを含むほ乳類と鳥類：無性生殖で増殖、シストの形成

　　　急性期の増殖盛んな急増虫体tachyzoiteとシスト内の緩増虫体bradyzoite

・病原、病因 phathogenesis

　・緩増虫体(bradyzoite)、接合子嚢(oocyst)

・感染経路

　1. オーシストの経口摂取

　2. 中間宿主の生肉中のシストの経口摂取

　3. 初感染妊婦からの経胎盤感染。既感染なら胎盤感染しないらしい(HIM.1306)

　(4)移植臓器、輸血。確率は低い(at low rate)(HIM.1306)

・病態

　1. 先天性トキソプラズマ症　congenital toxoplasmosis

　　　①網脈絡膜炎、 ②水頭症、 ③脳内石灰化、 ④精神・運動障害

　2. 後天性トキソプラズマ症　acquired toxoplasmosis

　　(1) 健常者

　　　・多くは不顕性感染。発熱、リンパ節腫脹、皮疹(rash)

　　　・(少数例)筋肉痛、暈疼痛、腹痛、斑状丘疹状皮疹(maculopapular rash)、脳脊髄炎、混乱(HIM.1308)

　　　・(まれ)肺炎、心筋炎、脳症、心膜炎、多発筋炎

　　　・網膜、脈絡叢に瘢痕や、脳に小さい炎症性の病変を残すことあり(CASES)。

　　　・急性感染の症状は数週間で消失　筋肉や中枢神経系に緩増虫体が残存

　　(2)HIV感染者、臓器移植例、がん化学療法例

　　　シスト内緩増虫体→急増虫体→播種性の多臓器感染

　　　AIDSでは、トキソプラズマ性脳炎が指標疾患 AIDS-defineing illness(CASES)

・治療

　(日本)アセチルスピラマイシン、ファンシダール(感染予防学 080521)

■トキソプラズマ脳炎 toxoplasmic encephalitis、トキソプラズマ脳症 cerebral toxoplasosis

・症状

　発熱、頭痛、混乱ｍ、痙攣、認知の障害、局所神経徴候(不全片麻痺、歯垢、脳神経損傷、視野欠損、感覚喪失)(CASES)

・画像検査

　(CT,MRI)多発性、両側性、ring-enhancing lesion、特に灰白質-白質境界、大脳基底核、脳幹、小脳が冒されやすい(CASES)

・鑑別診断(臨床症状・画像診断の所見で)

　リンパ腫、結核、転移性脳腫瘍(CASES)

・病歴と画像所見からの鑑別診断

　リンパ腫、結核、転移性腫瘍

？

このCTがcerebral toxoplasmosisに特徴的かは不明

□最後に残る疑問

　AIDSでWBC(leukocyte)の数はどうなるんだろう？？？AIDSの初診患者ではWBCが低い人が多いらしいし()、HIVはCD4+ T cellとmacrophageに感染して殺すから、これによってB cellは減るだろうし、CD8+ T cellも若干減少するだろうからWBCは減るんじゃないか？！好中球はAIDSとは関係ない？好中球は他の感染症に反応性に増加している？ちなみに、好酸球は寄生虫(原虫)の感染のために増える傾向にあるらしい(HIMのどこか)。

スルファジアジン

sulfadiazine

ピリメタミン

pyrimethamine

葉酸拮抗剤である。

サルファ剤と併用され、抗トキソプラズマ薬、抗ニューモシチス・カリニ薬として相乗的に働く。

ST合剤

SMX-TMP

スルファメトキサゾール・トリメトプリム合剤 sulfamethoxazole and trimethoprim mixture

□AIDSの定義(http://en.wikipedia.org/wiki/CDC_Classification_System_for_HIV_Infection_in_Adults_and_Adolescents)

• According to the US CDC definition, one has AIDS if he/she is infected with HIV and present with one of the following:

A CD4+ T-cell count below 200 cells/μl (or a CD4+ T-cell percentage of total lymphocytes of less than 14%).

or he/she has one of the following defining illnesses:

• 01. Candidiasis of bronchi, trachea, or lungs
• 02. Candidiasis esophageal
• 03. Cervical cancer (invasive)
• 04. Coccidioidomycosis, disseminated or extrapulmonary
• 05. Cryptococcosis, extrapulmonary
• 06. Cryptosporidiosis, chronic intestinal for longer than 1 month
• 07. Cytomegalovirus disease (other than liver, spleen or lymph nodes)
• 08. Encephalopathy (HIV-related)
• 09. Herpes simplex: chronic ulcer(s) (for more than 1 month); or bronchitis, pneumonitis, or esophagitis
• 10. Histoplasmosis, disseminated or extrapulmonary
• 11. Isosporiasis, chronic intestinal (for more than 1 month)
• 12. Kaposi's sarcoma
• 13. Lymphoma Burkitt's, immunoblastic or primary brain
• 14. Mycobacterium avium complex
• 15. Mycobacterium, other species, disseminated or extrapulmonary
• 16. Pneumocystis carinii pneumonia
• 17. Pneumonia (recurrent)
• 18. Progressive multifocal leukoencephalopathy
• 19. Salmonella septicemia (recurrent)
• 20. Toxoplasmosis of the brain
• 21. Tuberculosis
• 22. Wasting syndrome due to HIV

People who are not infected with HIV may also develop these conditions; this does not mean they have AIDS. However, when an individual presents laboratory evidence against HIV infection, a diagnosis of AIDS is ruled out unless the patient has not:

• undergone high-dose corticoid therapy or other immunosuppressive/cytotoxic therapy in the three months before the onset of the indicator disease
• OR been diagnosed with Hodgkin's disease, non-Hodgkin's lymphoma, lymphocytic leukemia, multiple myeloma, or any cancer of lymphoreticular or histiocytic tissue, or angioimmunoblastic lymphoadenopathy
• OR a genetic immunodeficiency syndrome atypical of HIV infection, such as one involving hypogamma globulinemia

AND

• the individual has had Pneumocystis carinii pneumonia
• OR one of the above defining illnesses AND a CD4+ T-cell count below 200 cells/μl (or a CD4+ T-cell percentage of total lymphocytes of less than 14%).

□AIDSのステージング

• Revised World Health Organization (WHO) Clinical Staging of HIV/AIDS For Adults and Adolescents (2005)

• http://en.wikipedia.org/wiki/WHO_Disease_Staging_System_for_HIV_Infection_and_Disease_in_Adults_and_Adolescents

■参考文献

HIM = Harrison's Principles of Internal Medicine 17th Edition

CASES = 100 Cases in Clinical Medicine Second edition

IMD = 内科診断学第2版

「100Cases 16」

　　[★]

☆case16　膝の痛み

■glossary

indigestion 消化障害、消化不良

■症例

80歳　男性

主訴：左膝の痛みと腫脹

現病歴：左膝の痛みを2日前から認めた。膝は発熱・腫脹しており、動かすと疼痛を生じる。時々胸焼けと消化不良が見られる。6ヶ月前のhealth checkで、高血圧(172/102mmHg)と血中クレアチニンが高い(正常高値)こと以外は正常といわれた。その4週間数回血圧を測定したが、高値が継続したため、2.5mg bendrofluamethizide(UK)/ベンドロフルメチアジドbendroflumethiazide(US)で治療を開始した。最近の血圧は138/84 mmHgであった。

喫煙歴：なし。

飲酒歴：一週間に平均4unit。

既往歴：股関節に中程度(mild)の変形性関節症

家族歴：特記なし

服薬歴：アセトアミノフェン(股関節の疼痛に対して)

身体所見 examination

　血圧 142/86mmHg。体温37.5℃。脈拍88/分。grade 2 hypertensive retinopathy(高血圧症性網膜症)。心血管系、呼吸器系に検査場異常なし。手にDIPにヘバーデン結節なし。

　左膝が発熱し腫脹している。関節内に液、patellar tap陽性。90℃以上膝関節を屈曲させると痛みを生じる。右の膝関節は正常に見える。

検査　investigation

　生化学：白血球増多、ESR上昇、尿素高値、グルコース高値

　単純X線：関節間隙やや狭小。それ以外に異常は認めない。

■problem list

　#1 左膝の痛み

　#2 胸焼け

　#3 消化不良

　#4 高血圧

　#5 クレアチニン正常高値

　#6 股関節の変形性リウマチ

　#7 高血圧性網膜症

■考え方

　・関節痛の鑑別診断を考える。

　・VINDICATEで考えてみてもよいでしょう。

　・関節痛の頻度としては　外傷＞慢性疾患(OAなど)＞膠原病＞脊椎疾患＞悪性腫瘍

■関節痛の鑑別疾患

DIF 282

V Vascular 血友病 hemophilia, 壊血病 scurvy, 無菌性骨壊死 aseptic bone necrosis (Osgood-Schlatter diseaseとか)

I Inflammatory 淋疾 gonorrhea, ライム病 lyme disease, 黄色ブドウ球菌 Staphylococcus, 連鎖球菌 Streptococcus, 結核 tuberculosis, 梅毒 syphilis, 風疹 rubella, 単純ヘルペス herpes simplex, HIV human immunodeficiency virus, サイトメガロウイルス cytomegalovirus

N Neoplastic disorders 骨原性肉腫 osteogenic sarcoma, 巨細胞腫 giant cell tumors

D Degenerative disorders degenerative joint disease or 変形性関節症 osteoarthritis

I Intoxication 痛風 gout (uric acid), 偽痛風 pseudogout (calcium pyrophosphate), ループス症候群 lupus syndrome of hydralazine (Apresoline) and procainamide, gout syndrome of diuretics

C Congenital and acquired malformations bring to mind the joint deformities of tabes dorsalis and syringomyelia and congenital dislocation of the hip. Alkaptonuria is also considered here.

A Autoimmune indicates (多い)関節リウマチ RA (可能性)血清病 serum sickness, 全身性エリテマトーデス lupus erythematosus, リウマチ熱 rheumatic fever, ライター症候群 Reiter syndrome, 潰瘍性大腸炎 ulcerative colitis, クローン病=限局性回腸炎 regional ileitis, 乾癬性関節炎 psoriatic arthritis (老人であり得る)リウマチ性多発筋痛症 polymyalgia rheumatica

T Trauma 外傷性滑膜炎 traumatic synovitis, tear or rupture of the collateral or cruciate ligaments, 亜脱臼 subluxation or laceration of the meniscus (semilunar cartilage), 脱臼 dislocation of the joint or patella, a 捻挫　sprain of the joint, and fracture of the bones of the joint.

E Endcrine 先端肥大症 acromegaly, 閉経 menopause, 糖尿病 diabetes mellitus

■答え

　骨格筋系-関節炎-単関節炎-急性単関節炎

　痛風　尿酸　→　発熱、ESR↑、白血球↑

　偽痛風　ピロリン酸カルシウム

　高齢女性でチアジド系利尿薬の使用により痛風が誘発されやすい。特に腎機能低下、糖尿病の人はこのリスクが高まる。

■(BSTからの知識「)循環器領域での利尿薬

・心不全の治療において、循環血漿量を減らし、心臓の前負荷を軽減する。

・利尿薬は高尿酸血症を起こす。(けど、心不全の治療において高尿酸血症になったからといって痛風を発症している患者はみたことない)

・電解質異常を起こしやすいので、血液生化学の検査でモニタして注意する。たとえば低Kで不整脈のリスクが高まる。

・チアジド系の利尿薬は血糖を上げるし、尿酸を上げる

・長期の使用で腎機能を低下させる

■initial plan

　Dx 1. 関節液の吸引：関節液の一般検査、生化学検査、培養検査、

　　　　・白血球が増加していれば急性炎症性であることを示す。

　　　　・偏光顕微鏡で関節液を検鏡する。

　　　　　・尿酸の結晶：針状結晶。negatively birefringent

　　　　　・ピロリン酸カルシウムの結晶：positively birefringent

　Tx 1. 関節液の吸引：炎症が軽度改善

　　 2. NSAIDによる疼痛管理

　　 3. PPI：NSAID潰瘍を予防するため

　　 4. ACE inhibitorの導入

「単一性」

　　[★]

英

simplex、simple

関

単一、単式、単純、単純性、簡素、シンプル、簡便

「単式」

　　[★]

英

simplex

関

単一性

「herpes simplex virus protein Vmw65」

　　[★]

単純ヘルペスウイルスタンパク質Vmw65

関

VP16 protein

「herpes simplex virus vaccine」

　　[★] 単純ヘルペスワクチン

「herpes simplex genitalis」

　　[★]

外陰部単純ヘルペス

「simple」

　　[★]

• adj.

• (比較級simpler-最上級simplest)単純な、簡便な、シンプルな、簡素な、単一の、単一性の、単純性の

関

convenient、mono、parsimonious、plain、simplex、simplicity、simply、single、unity