Editing Simplex (section)

== Geometric properties ==

=== Volume ===
The [[volume]] of an {{mvar|n}}-simplex in {{mvar|n}}-dimensional space with vertices {{math|(''v''<sub>0</sub>, ..., ''v''<sub>''n''</sub>)}} is
: <math>
\mathrm{Volume} = \frac{1}{n!} \left|\det
\begin{pmatrix}
v_1-v_0 && v_2-v_0 && \cdots && v_n-v_0
\end{pmatrix}\right|
</math>

where each column of the {{math|''n'' × ''n''}} [[determinant]] is a [[vector (geometry)|vector]] that points from vertex {{math|''v''{{sub|0}}}} to another vertex {{math|''v''{{sub|''k''}}}}.<ref>A derivation of a very similar formula can be found in {{cite journal | last1 = Stein | first1 = P. | year = 1966 | title = A Note on the Volume of a Simplex | journal = American Mathematical Monthly | volume = 73 | issue = 3 | pages = 299–301 | jstor = 2315353 | doi = 10.2307/2315353 }}</ref> This formula is particularly useful when <math>v_0</math> is the origin.

The expression
: <math>
\mathrm{Volume} = \frac{1}{n!} \det\left[
\begin{pmatrix}
v_1^\text{T}-v_0^\text{T} \\ v_2^\text{T}-v_0^\text{T} \\ \vdots \\ v_n^\text{T}-v_0^\text{T}
\end{pmatrix}
\begin{pmatrix}
v_1-v_0 & v_2-v_0 & \cdots & v_n-v_0
\end{pmatrix}
\right]^{1/2}
</math>
employs a [[Gram determinant]] and works even when the {{mvar|n}}-simplex's vertices are in a Euclidean space with more than {{mvar|n}} dimensions, e.g., a triangle in <math>\mathbf{R}^3</math>. 

A more symmetric way to compute the volume of an {{mvar|n}}-simplex in <math>\mathbf{R}^n</math> is
: <math>
\mathrm{Volume} = {1\over n!} \left|\det
\begin{pmatrix}
v_0 & v_1 & \cdots & v_n \\
1   & 1   & \cdots & 1
\end{pmatrix}\right|.
</math>

Another common way of computing the volume of the simplex is via the [[Cayley–Menger determinant]], which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.<ref>{{mathworld|title=Cayley-Menger Determinant |author=Colins, Karen D. }}</ref>

Without the {{math|1/''n''!}} it is the formula for the volume of an {{mvar|n}}-[[parallelepiped#Parallelotope|parallelotope]]. 
This can be understood as follows: Assume that {{mvar|P}} is an {{mvar|n}}-parallelotope constructed on a basis <math>(v_0, e_1, \ldots, e_n)</math> of <math>\mathbf{R}^n</math>.
Given a [[permutation]] <math>\sigma</math> of <math>\{1,2,\ldots, n\}</math>, call a list of vertices <math>v_0,\ v_1, \ldots, v_n</math> a {{mvar|n}}-path if 
: <math>v_1 = v_0 + e_{\sigma(1)},\ v_2 = v_1 + e_{\sigma(2)},\ldots, v_n = v_{n-1}+e_{\sigma(n)}</math> 
(so there are {{math|''n''!}} {{mvar|n}}-paths and <math>v_n</math> does not depend on the permutation). The following assertions hold:

If {{mvar|P}} is the unit {{mvar|n}}-hypercube, then the union of the {{mvar|n}}-simplexes formed by the convex hull of each {{mvar|n}}-path is {{mvar|P}}, and these simplexes are congruent and pairwise non-overlapping.<ref>Every {{mvar|n}}-path corresponding to a permutation <math>\scriptstyle \sigma</math> is the image of the {{mvar|n}}-path <math>\scriptstyle v_0,\ v_0+e_1,\ v_0+e_1+e_2,\ldots v_0+e_1+\cdots + e_n</math> by the affine isometry that sends <math>\scriptstyle v_0</math> to <math>\scriptstyle v_0</math>, and whose linear part matches <math>\scriptstyle e_i</math> to <math>\scriptstyle e_{\sigma(i)}</math> for all&nbsp;{{mvar|i}}. hence every two {{mvar|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 {{mvar|n}}-path <math>\scriptstyle v_0,\ v_0+e_{\sigma(1)},\ v_0+e_{\sigma(1)}+e_{\sigma(2)}\ldots v_0+e_{\sigma(1)}+\cdots + e_{\sigma(n)}</math> is the set of points <math>\scriptstyle v_0 + (x_1+\cdots +x_n) e_{\sigma(1)} + \cdots + (x_{n-1}+x_n) e_{\sigma(n-1)} + x_n e_{\sigma(n)}</math>, with <math>\scriptstyle 0< x_i < 1</math> and <math>\scriptstyle x_1+\cdots + x_n < 1.</math> 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 {{mvar|n}}-hypercube follows as well, replacing the strict inequalities above by "<math>\scriptstyle \leq</math>". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.</ref> In particular, the volume of such a simplex is
: <math> \frac{\operatorname{Vol}(P)}{n!} = \frac 1 {n!}.</math>

If {{mvar|P}} is a general parallelotope, the same assertions hold except that it is no longer true, in dimension&nbsp;> 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the {{mvar|n}}-parallelotope is the image of the unit {{mvar|n}}-hypercube by the [[linear isomorphism]] that sends the canonical basis of <math>\mathbf{R}^n</math> to <math>e_1,\ldots, e_n</math>. As previously, this implies that the volume of a simplex coming from a {{mvar|n}}-path is:
: <math> \frac{\operatorname{Vol}(P)}{n!} = \frac{\det(e_1, \ldots, e_n)}{n!}.</math>

Conversely, given an {{mvar|n}}-simplex <math>(v_0,\ v_1,\ v_2,\ldots v_n)</math> of <math>\mathbf R^n</math>, it can be supposed that the vectors <math>e_1 = v_1-v_0,\ e_2 = v_2-v_1,\ldots e_n=v_n-v_{n-1}</math> form a basis of <math>\mathbf R^n</math>. Considering the parallelotope constructed from <math>v_0</math> and <math>e_1,\ldots, e_n</math>, one sees that the previous formula is valid for every simplex.

Finally, the formula at the beginning of this section is obtained by observing that 
: <math>\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}).</math>

From this formula, it follows immediately that the volume under a standard {{mvar|n}}-simplex (i.e. between the origin and the simplex in {{math|'''R'''<sup>''n''+1</sup>}}) is
: <math>{1 \over (n+1)!}</math>

The volume of a regular {{mvar|n}}-simplex with unit side length is
: <math>\frac{\sqrt{n+1}}{n!\sqrt{2^n}}</math>
as can be seen by multiplying the previous formula by {{math|''x''<sup>''n''+1</sup>}}, to get the volume under the {{mvar|n}}-simplex as a function of its vertex distance {{mvar|x}} from the origin, differentiating with respect to {{mvar|x}}, at <math>x=1/\sqrt{2}</math>&nbsp; (where the {{mvar|n}}-simplex side length is 1), and normalizing by the length <math>dx/\sqrt{n+1}</math> of the increment, <math>(dx/(n+1),\ldots, dx/(n+1))</math>, along the normal vector.

=== Dihedral angles of the regular ''n''-simplex ===
Any two {{math|(''n'' − 1)}}-dimensional faces of a regular {{mvar|n}}-dimensional simplex are themselves regular {{math|(''n'' − 1)}}-dimensional simplices, and they have the same [[dihedral angle]] of {{math|cos<sup>−1</sup>(1/''n'')}}.<ref>{{cite journal | journal =American Mathematical Monthly | volume = 109 | issue = 8 | date = October 2002 | pages = 756–8 | title = An Elementary Calculation of the Dihedral Angle of the Regular {{mvar|n}}-Simplex | first1 = Harold R. | last1 = Parks | author-link = Harold R. Parks |first2 = Dean C. |last2=Wills | jstor = 3072403 | doi=10.2307/3072403}}</ref><ref>{{cite thesis |type=PhD | publisher = Oregon State University | date = June 2009 | title = Connections between combinatorics of permutations and algorithms and geometry |first1= Harold R. |last2=Parks |first2 = Dean C. |last1=Wills | url = http://ir.library.oregonstate.edu/xmlui/handle/1957/11929 |hdl=1957/11929}}</ref> 

This can be seen by noting that the center of the standard simplex is <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math>, and the centers of its faces are coordinate permutations of <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math>. Then, by symmetry, the vector pointing from <math display="inline">\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)</math> to <math display="inline">\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)</math> is perpendicular to the faces. So the vectors normal to the faces are permutations of <math>(-n, 1, \dots, 1)</math>, from which the dihedral angles are calculated.

=== Simplices with an "orthogonal corner" ===
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent [[Face (geometry)|faces]] are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an {{mvar|n}}-dimensional version of the [[Pythagorean theorem]]: The sum of the squared {{math|(''n'' − 1)}}-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared {{math|(''n'' − 1)}}-dimensional volume of the facet opposite of the orthogonal corner.
: <math> \sum_{k=1}^n |A_k|^2 = |A_0|^2 </math>
where <math> A_1 \ldots A_n </math> are facets being pairwise orthogonal to each other but not orthogonal to <math>A_0</math>, which is the facet opposite the orthogonal corner.<ref>{{cite journal |last1=Donchian |first1=P. S. |last2=Coxeter |first2=H. S. M. |date=July 1935 |title=1142. An n-dimensional extension of Pythagoras' Theorem |journal=The Mathematical Gazette |volume=19 |issue=234 |pages=206 |doi=10.2307/3605876|jstor=3605876 |s2cid=125391795 }}</ref>

For a 2-simplex, the theorem is the [[Pythagorean theorem]] for triangles with a right angle and for a 3-simplex it is [[de Gua's theorem]] for a tetrahedron with an orthogonal corner.

=== Relation to the (''n'' + 1)-hypercube ===
The [[Hasse diagram]] of the face lattice of an {{mvar|n}}-simplex is isomorphic to the graph of the {{math|(''n'' + 1)}}-[[hypercube]]'s edges, with the hypercube's vertices mapping to each of the {{mvar|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 {{mvar|n}}-simplex is also the [[vertex figure]] of the {{math|(''n'' + 1)}}-hypercube. It is also the [[Facet (geometry)|facet]] of the {{math|(''n'' + 1)}}-[[orthoplex]].

=== Topology ===
[[Topology|Topologically]], an {{mvar|n}}-simplex is [[topologically equivalent|equivalent]] to an [[ball (mathematics)|{{mvar|n}}-ball]]. Every {{mvar|n}}-simplex is an {{mvar|n}}-dimensional [[manifold with corners]].

=== Probability ===
{{Main|Categorical distribution}}

In probability theory, the points of the standard {{mvar|n}}-simplex in {{math|(''n'' + 1)}}-space form the space of possible probability distributions on a finite set consisting of {{math|''n'' + 1}} possible outcomes. The correspondence is as follows: For each distribution described as an ordered {{math|(''n'' + 1)}}-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose [[barycentric coordinates]] are precisely those probabilities. That is, the {{mvar|k}}th vertex of the simplex is assigned to have the {{mvar|k}}th probability of the {{math|(''n'' + 1)}}-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.

=== Aitchison geometry ===
{{Main|Aitchison geometry}}

Aitchinson geometry is a natural way to construct an [[inner product space]] from the standard simplex <math>\Delta^{D-1}</math>. It defines the following operations on simplices and real numbers:

; Perturbation (addition)

:: <math> x \oplus y = \left[\frac{x_1 y_1}{\sum_{i=1}^D x_i y_i},\frac{x_2 y_2}{\sum_{i=1}^D x_i y_i}, \dots, \frac{x_D y_D}{\sum_{i=1}^D x_i y_i}\right]  \qquad \forall x, y \in \Delta^{D-1}
</math>

; Powering (scalar multiplication)

:: <math> \alpha \odot x = \left[\frac{x_1^\alpha}{\sum_{i=1}^D x_i^\alpha},\frac{x_2^\alpha}{\sum_{i=1}^D x_i^\alpha}, \ldots,\frac{x_D^\alpha}{\sum_{i=1}^D x_i^\alpha} \right]  \qquad \forall x \in \Delta^{D-1}, \; \alpha \in \mathbb{R}
</math>

; Inner product

:: <math> \langle x, y \rangle = \frac{1}{2D} 
\sum_{i=1}^D 
\sum_{j=1}^D
\log \frac{x_i}{x_j} 
\log \frac{y_i}{y_j} 
\qquad \forall x, y \in \Delta^{D-1}</math>

=== Compounds ===
Since all simplices are self-dual, they can form a series of compounds;
* Two triangles form a [[hexagram]] {6/2}.
* Two tetrahedra form a [[compound of two tetrahedra]] or [[stellated octahedron|stella octangula]].
* Two 5-cells form a [[compound of two 5-cells]] in four dimensions.