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[Other uses] [Multi-dimensional generalization of triangle] alt=The four
simplexes that can be fully represented in 3D space.
In geometry, a SIMPLEX (plural: SIMPLEXES or SIMPLICES) is a generalization of
the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex
is so-named because it represents the simplest possible polytope in any given
dimension. For example,
- a 0-dimensional simplex is a point ,
- a 1-dimensional simplex is a line segment ,
- a 2-dimensional simplex is a triangle ,
- a 3-dimensional simplex is a tetrahedron , and
- a 4-dimensional simplex is a 5-cell .
Specifically, a [K] -SIMPLEX is a [k]-dimensional polytope that is the convex
hull of its [_k_ + 1] vertices. More formally, suppose the [_k_ + 1] points
<math>u_0, \dots, u_k</math> are affinely independent, which means that the
[k] vectors <math>u_1 - u_0,\dots, u_k-u_0</math> are linearly independent.
Then, the simplex determined by them is the set of points <math
display="block"> C = \left\{\theta_0 u_0 + \dots +\theta_k u_k ~\Bigg|~
\sum_{i=0}^{k} \theta_i=1 \mbox{ and } \theta_i \ge 0 \mbox{ for } i = 0,
\dots, k\right\}.</math>
A REGULAR SIMPLEX[1] is a simplex that is also a regular polytope. A regular
[k]-simplex may be constructed from a regular [(_k_ − 1)]-simplex by
connecting a new vertex to all original vertices by the common edge length.
The STANDARD SIMPLEX or PROBABILITY SIMPLEX[2] is the [(_k_ − 1)]-dimensional
simplex whose vertices are the [k] standard unit vectors in
<math>\mathbf{R}^k</math>, or in other words <math display="block">\left\{x
\in \mathbf{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0,
\dots, k-1 \right\}.</math>
In topology and combinatorics, it is common to "glue together" simplices to
form a simplicial complex.
The geometric simplex and simplicial complex should not be confused with the
abstract simplicial complex, in which a simplex is simply a finite set and the
complex is a family of such sets that is closed under taking subsets.
** History
The concept of a simplex was known to William Kingdon Clifford, who wrote
about these shapes in 1886 but called them "prime confines". Henri Poincaré,
writing about algebraic topology in 1900, called them "generalized
tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with
the Latin superlative _simplicissimum_ ("simplest") and then with the same
Latin adjective in the normal form _simplex_ ("simple").[3]
The REGULAR SIMPLEX family is the first of three regular polytope families,
labeled by Donald Coxeter as [_α_ <sub> _n_ </sub>], the other two being the
cross-polytope family, labeled as [_β_ <sub> _n_ </sub>], and the hypercubes,
labeled as [_γ_ <sub> _n_ </sub>]. A fourth family, the tessellation of [n]
-dimensional space by infinitely many hypercubes, he labeled as [_δ_ <sub> _n_
</sub>].[Coxeter]
** Elements
<!-- This section is linked from Simplicial complex --> 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
<math>\tbinom{n+1}{m+1}</math>.[Coxeter] 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.
The extended f-vector for an [n]-simplex can be computed by [( 1 , 1 ) <sup>
_n_ +1 </sup>], like the coefficients of polynomial products. For example, a
7-simplex is ( 1, 1)<sup>8</sup> = ( 1,2, 1)<sup>4</sup> = ( 1,4,6,4,
1)<sup>2</sup> = ( 1,8,28,56,70,56,28,8, 1).
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.
{| class="wikitable" |+ [n]-Simplex elements[4] |- ! [Δ<sup> _n_ </sup>] !
Name !Schläfli<br />Coxeter ! 0-<br />faces<br /><small>(vertices)</small> !
1-<br />faces<br /><small>(edges)</small> ! 2-<br />faces<br
/><small>(faces)</small> ! 3-<br />faces<br /><small>(cells)</small> ! 4-<br
/>faces<br /><small> </small> ! 5-<br />faces<br /><small> </small> ! 6-<br
/>faces<br /><small> </small> ! 7-<br />faces<br /><small> </small> ! 8-<br
/>faces<br /><small> </small> ! 9-<br />faces<br /><small> </small> ! 10-<br
/>faces<br /><small> </small> ! SUM<br />= 2<sup>_n_+1</sup> − 1 |- !
Δ<sup>0</sup> | 0-simplex<br />(point) | ( )<br />[node] | 1 | | | | | | | | |
| | 1 |- ! Δ<sup>1</sup> | 1-simplex<br />(line segment) |{ } = ( ) ∨ ( ) =
2⋅( )<br />[node_1] | 2 | 1 | | | | | | | | | | 3 |- ! Δ<sup>2</sup> |
2-simplex<br />(triangle) |{3} = 3⋅( )<br />[node_1] | 3 | 3 | 1 | | | | | | |
| | 7 |- ! Δ<sup>3</sup> | 3-simplex<br />(tetrahedron) |{3,3} = 4⋅( )<br
/>[node_1] | 4 | 6 | 4 | 1 | | | | | | | | 15 |- ! Δ<sup>4</sup> |
4-simplex<br />(5-cell) |{3<sup>3</sup>} = 5⋅( )<br />[node_1] | 5 | 10 | 10 |
5 | 1 | | | | | | | 31 |- ! Δ<sup>5</sup> | 5-simplex |{3<sup>4</sup>} = 6⋅(
)<br />[node_1] | 6 | 15 | 20 | 15 | 6 | 1 | | | | | | 63 |- ! Δ<sup>6</sup> |
6-simplex |{3<sup>5</sup>} = 7⋅( )<br />[node_1] | 7 | 21 | 35 | 35 | 21 | 7 |
1 | | | | | 127 |- ! Δ<sup>7</sup> | 7-simplex |{3<sup>6</sup>} = 8⋅( )<br
/>[node_1] | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | | | | 255 |- ! Δ<sup>8</sup>
| 8-simplex |{3<sup>7</sup>} = 9⋅( )<br />[node_1] | 9 | 36 | 84 | 126 | 126 |
84 | 36 | 9 | 1 | | | 511 |- ! Δ<sup>9</sup> | 9-simplex |{3<sup>8</sup>} =
10⋅( )<br />[node_1] | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | |
1023 |- ! Δ<sup>10</sup> | 10-simplex |{3<sup>9</sup>} = 11⋅( )<br />[node_1]
| 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | 2047 |}
An [n]-simplex is the polytope with the fewest vertices that requires [n]
dimensions. Consider a line segment _AB_ as a shape in a 1-dimensional space
(the 1-dimensional space is the line in which the segment lies). One can
place a new point [C] somewhere off the line. The new shape, triangle _ABC_,
requires two dimensions; it cannot fit in the original 1-dimensional space.
The triangle is the 2-simplex, a simple shape that requires two dimensions.
Consider a triangle _ABC_, a shape in a 2-dimensional space (the plane in
which the triangle resides). One can place a new point [D] somewhere off the
plane. The new shape, tetrahedron _ABCD_, requires three dimensions; it
cannot fit in the original 2-dimensional space. The tetrahedron is the
3-simplex, a simple shape that requires three dimensions. Consider
tetrahedron _ABCD_, a shape in a 3-dimensional space (the 3-space in which the
tetrahedron lies). One can place a new point [E] somewhere outside the
3-space. The new shape _ABCDE_, called a 5-cell, requires four dimensions and
is called the 4-simplex; it cannot fit in the original 3-dimensional space.
(It also cannot be visualized easily.) This idea can be generalized, that is,
adding a single new point outside the currently occupied space, which requires
going to the next higher dimension to hold the new shape. This idea can also
be worked backward: the line segment we started with is a simple shape that
requires a 1-dimensional space to hold it; the line segment is the 1-simplex.
The line segment itself was formed by starting with a single point in
0-dimensional space (this initial point is the 0-simplex) and adding a second
point, which required the increase to 1-dimensional space.
More formally, 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 the
join of two points: [1=( ) ∨ ( ) = 2 ⋅ ( )]. A general 2-simplex (scalene
triangle) is the join of three 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 two 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.
{| |- | |- | |}
In some conventions,[5] the empty set is defined to be a (−1)-simplex. The
definition of the simplex above still makes sense if [1= _n_ = −1]. This
convention is more common in applications to algebraic topology (such as
simplicial homology) than to the study of polytopes. [clear]
** 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. {|
class=wikitable |- align=center |100px<br />1 |100px<br />2 |100px<br />3
|100px<br />4 |100px<br />5 |- align=center |100px<br />6 |100px<br />7
|100px<br />8 |100px<br />9 |100px<br />10 |- align=center |100px<br />11
|100px<br />12 |100px<br />13 |100px<br />14 |100px<br />15 |- align=center
|100px<br />16 |100px<br />17 |100px<br />18 |100px<br />19 |100px<br />20 |}
** Standard simplex
The STANDARD [N] -SIMPLEX (or UNIT [N] -SIMPLEX) is the subset of [ R <sup>
_n_ +1 </sup>] given by
<math> \Delta^n = \left\{(t_0,\dots,t_n)\in\mathbf{R}^{n+1} ~\Bigg | ~ \sum_{i = 0}^n t_i = 1 \text{ and } t_i \ge 0 \text{ for } i = 0, \ldots, n\right\} </math> .
The simplex [Δ<sup> _n_ </sup>] lies in the affine hyperplane obtained by
removing the restriction [_t_ <sub> _i_ </sub> ≥ 0] in the above definition.
The [_n_ + 1] vertices of the standard [n]-simplex are the points [_e_ <sub>
_i_ </sub> ∈ R <sup> _n_ +1 </sup>], where
[1= _e_ <sub> 0 </sub> = (1, 0, 0, ..., 0),]
[1= _e_ <sub> 1 </sub> = (0, 1, 0, ..., 0),]
⋮
[1= _e_ <sub> _n_ </sub> = (0, 0, 0, ..., 1)] .
A _standard simplex_ is an example of a 0/1-polytope, with all coordinates as
0 or 1. It can also be seen one facet of a regular [(_n_ + 1)]-orthoplex.
There is a canonical map from the standard [n]-simplex to an arbitrary
[n]-simplex with vertices ([_v_ <sub> 0 </sub>], ..., [_v_ <sub> _n_ </sub>])
given by
<math> (t_0,\ldots,t_n) \mapsto \sum_{i = 0}^n t_i v_i </math>
The coefficients [_t_ <sub> _i_ </sub>] 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
<math>(n-1)</math>-simplex (with [n] vertices) onto any polytope with [n]
vertices, given by the same equation (modifying indexing):
<math> (t_1,\ldots,t_n) \mapsto \sum_{i = 1}^n t_i v_i </math>
These are known as generalized barycentric coordinates, and express every
polytope as the _image_ of a simplex: <math>\Delta^{n-1} \twoheadrightarrow
P.</math>
A commonly used function from [ R <sup> _n_ </sup>] to the interior of the
standard <math>(n-1)</math>-simplex is the softmax function, or normalized
exponential function; this generalizes the standard logistic function.
*** Examples
- Δ<sup> 0 </sup> is the point [1] in [ R <sup> 1 </sup>] .
- Δ<sup> 1 </sup> is the line segment joining [(1, 0)] and [(0, 1)] in [ R <sup> 2 </sup>] .
- Δ<sup> 2 </sup> is the equilateral triangle with vertices [(1, 0, 0)] , [(0, 1, 0)] and [(0, 0, 1)] in [ R <sup> 3 </sup>] .
- Δ<sup> 3 </sup> is the regular tetrahedron with vertices [(1, 0, 0, 0)] , [(0, 1, 0, 0)] , [(0, 0, 1, 0)] and [(0, 0, 0, 1)] in [ R <sup> 4 </sup>] .
- Δ<sup> 4 </sup> is the regular 5-cell with vertices [(1, 0, 0, 0, 0)] , [(0, 1, 0, 0, 0)] , [(0, 0, 1, 0, 0)] , [(0, 0, 0, 1, 0)] and [(0, 0, 0, 0, 1)] in [ R <sup> 5 </sup>] .
*** Increasing coordinates
An alternative coordinate system is given by taking the indefinite sum:
<math>
\begin{align} 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\\ &\;\;\vdots\\ s_n &= s_{n-1} +
t_{n-1} = t_0 + t_1 + \cdots + t_{n-1}\\ s_{n+1} &= s_n + t_n = t_0 + t_1 +
\cdots + t_n = 1 \end{align} </math> This yields the alternative presentation
by _order,_ namely as nondecreasing [n]-tuples between 0 and 1:
<math> \Delta_*^n = \left\{(s_1,\ldots,s_n)\in\mathbf{R}^n\mid 0 = s_0 \leq s_1 \leq s_2 \leq \dots \leq s_n \leq s_{n+1} = 1 \right\}. </math>
Geometrically, this is an [n]-dimensional subset of <math>\mathbf{R}^n</math>
(maximal dimension, codimension 0) rather than of
<math>\mathbf{R}^{n+1}</math> (codimension 1). The facets, which on the
standard simplex correspond to one coordinate vanishing, <math>t_i=0,</math>
here correspond to successive coordinates being equal,
<math>s_i=s_{i+1},</math> 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
<math>n!</math> 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_ <sup> 2 </sup> /2], [_x_ <sup> 3 </sup> /3!], ..., [_x_ <sup> _n_
</sup> / _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], possibly with coordinates
that are negative or in excess of 1, the closest point [t] on the simplex has
coordinates
<math> t_i= \max\{p_i+\Delta\, ,0\}, </math>
where <math>\Delta</math> is chosen such that <math
display="inline">\sum_i\max\{p_i+\Delta\, ,0\}=1.</math>
<math>\Delta</math> can be easily calculated from sorting the coordinates of
[p].[6] The sorting approach takes <math>O( n \log n)</math> complexity, which
can be improved to [O( _n_ )] complexity via median-finding algorithms.[7]
Projecting onto the simplex is computationally similar to projecting onto the
<math>\ell_1</math> ball. Also see Integer programming.
*** 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:
<math> \Delta_c^n = \left\{(t_1,\ldots,t_n)\in\mathbf{R}^n ~\Bigg | ~ \sum_{i = 1}^n t_i \leq 1 \text{ and } t_i \ge 0 \text{ for all } i \right\}. </math>
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 a regular [n] -dimensional simplex in R <sup> _n_ </sup>
One way to write down a regular [n]-simplex in [ R <sup> _n_ </sup>] is to
choose two points to be the first two vertices, choose a third point to make
an equilateral triangle, choose a fourth point to make a regular tetrahedron,
and so on. Each step requires satisfying equations that ensure that each
newly chosen vertex, together with the previously chosen vertices, forms a
regular simplex. There are several sets of equations that can be written down
and used for this purpose. These include the equality of all the distances
between vertices; the equality of all the distances from vertices to the
center of the simplex; the fact that the angle subtended through the new
vertex by any two previously chosen vertices is <math>\pi/3</math>; and the
fact that the angle subtended through the center of the simplex by any two
vertices is <math>\arccos(-1/n)</math>.
It is also possible to directly write down a particular regular [n]-simplex in
[ R <sup> _n_ </sup>] which can then be translated, rotated, and scaled as
desired. One way to do this is as follows. Denote the basis vectors of [ R
<sup> _n_ </sup>] by [ E <sub> 1 </sub>] through [ E <sub> _n_ </sub>]. Begin
with the standard [(_n_ − 1)]-simplex which is the convex hull of the basis
vectors. By adding an additional vertex, these become a face of a regular
[n]-simplex. The additional vertex must lie on the line perpendicular to the
barycenter of the standard simplex, so it has the form [(_α_ / _n_ , ..., _α_
/ _n_ )] for some real number [α]. Since the squared distance between two
basis vectors is 2, in order for the additional vertex to form a regular
[n]-simplex, the squared distance between it and any of the basis vectors must
also be 2. This yields a quadratic equation for [α]. Solving this equation
shows that there are two choices for the additional vertex:
<math> \frac{1}{n} \left(1 \pm \sqrt{n + 1} \right) \cdot (1, \dots, 1). </math>
Either of these, together with the standard basis vectors, yields a regular
[n]-simplex.
The above regular [n]-simplex is not centered on the origin. It can be
translated to the origin by subtracting the mean of its vertices. By
rescaling, it can be given unit side length. This results in the simplex
whose vertices are:
<math> \frac{1}{\sqrt{2 }} \mathbf{e}_i - \frac{1}{n\sqrt{2 }} \bigg(1 \pm \frac{1}{\sqrt{n + 1 }} \bigg) \cdot (1, \dots, 1), </math>
for <math>1 \le i \le n</math>, and
<math> \pm\frac{1}{\sqrt{2(n + 1) }} \cdot (1, \dots, 1). </math>
Note that there are two sets of vertices described here. One set uses
<math>+</math> in each calculation. The other set uses <math>-</math> in each
calculation.
This simplex is inscribed in a hypersphere of radius <math>\sqrt{n/(2(n +
1))}</math>.
A different rescaling produces a simplex that is inscribed in a unit
hypersphere. When this is done, its vertices are
<math> \sqrt{1 + n^{-1 }} \cdot\mathbf{e}_i - n^{-3/2}(\sqrt{n + 1} \pm 1) \cdot (1, \dots, 1), </math>
where <math>1 \le i \le n</math>, and
<math> \pm n^{-1/2} \cdot (1, \dots, 1). </math>
The side length of this simplex is <math display="inline">\sqrt{2(n +
1)/n}</math>.
A highly symmetric way to construct a regular [n]-simplex is to use a
representation of the cyclic group [ Z <sub> _n_ +1 </sub>] by orthogonal
matrices. This is an [_n_ × _n_] orthogonal matrix [Q] such that [1= _Q_
<sup> _n_ +1 </sup> = _I_] is the identity matrix, but no lower power of [Q]
is. Applying powers of this matrix to an appropriate vector [ V] will produce
the vertices of a regular [n]-simplex. To carry this out, first observe that
for any orthogonal matrix [Q], there is a choice of basis in which [Q] is a
block diagonal matrix
<math> Q = \operatorname{diag}(Q_1, Q_2, \dots, Q_k), </math>
where each [_Q_ <sub> _i_ </sub>] is orthogonal and either [2 × 2] or [1 × 1].
In order for [Q] to have order [_n_ + 1], all of these matrices must have
order dividing [_n_ + 1]. Therefore each [_Q_ <sub> _i_ </sub>] is either a
[1 × 1] matrix whose only entry is [1] or, if [n] is odd, [−1]; or it is a [2
× 2] matrix of the form
<math> \begin{pmatrix}
\cos \frac{2\pi\omega_i}{n + 1} & -\sin \frac{2\pi\omega_i}{n + 1} \\ \sin
\frac{2\pi\omega_i}{n + 1} & \cos \frac{2\pi\omega_i}{n + 1}
\end{pmatrix},</math> where each [_ω_ <sub> _i_ </sub>] is an integer between
zero and [n] inclusive. A sufficient condition for the orbit of a point to be
a regular simplex is that the matrices [_Q_ <sub> _i_ </sub>] form a basis for
the non-trivial irreducible real representations of [ Z <sub> _n_ +1 </sub>],
and the vector being rotated is not stabilized by any of them.
In practical terms, for [n] even this means that every matrix [_Q_ <sub> _i_
</sub>] is [2 × 2], there is an equality of sets
<math> \{\omega_1, n + 1 - \omega_1, \dots, \omega_{n/2}, n + 1 - \omega_{n/2}\} = \{1, \dots, n\}, </math>
and, for every [_Q_ <sub> _i_ </sub>], the entries of [ V] upon which [_Q_
<sub> _i_ </sub>] acts are not both zero. For example, when [1= _n_ = 4], one
possible matrix is
<math> \begin{pmatrix}
\cos(2\pi/5) & -\sin(2\pi/5) & 0 & 0 \\ \sin(2\pi/5) & \cos(2\pi/5) & 0 & 0 \\
0 & 0 & \cos(4\pi/5) & -\sin(4\pi/5) \\ 0 & 0 & \sin(4\pi/5) & \cos(4\pi/5)
\end{pmatrix}.</math> Applying this to the vector [(1, 0, 1, 0)] results in
the simplex whose vertices are
<math>
\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} \cos(2\pi/5)
\\ \sin(2\pi/5) \\ \cos(4\pi/5) \\ \sin(4\pi/5) \end{pmatrix}, \begin{pmatrix}
\cos(4\pi/5) \\ \sin(4\pi/5) \\ \cos(8\pi/5) \\ \sin(8\pi/5) \end{pmatrix},
\begin{pmatrix} \cos(6\pi/5) \\ \sin(6\pi/5) \\ \cos(2\pi/5) \\ \sin(2\pi/5)
\end{pmatrix}, \begin{pmatrix} \cos(8\pi/5) \\ \sin(8\pi/5) \\ \cos(6\pi/5) \\
\sin(6\pi/5) \end{pmatrix}, </math> each of which has distance √5 from the
others. When [n] is odd, the condition means that exactly one of the diagonal
blocks is [1 × 1], equal to [−1], and acts upon a non-zero entry of [ V];
while the remaining diagonal blocks, say [_Q_ <sub> 1 </sub> , ..., _Q_ <sub>
( _n_ − 1) / 2 </sub>], are [2 × 2], there is an equality of sets
<math> \left\{\omega_1, -\omega_1, \dots, \omega_{(n-1)/2}, -\omega_{(n-1)/2}\right\} = \left\{1, \dots, (n-1)/2, (n+3)/2, \dots, n \right\}, </math>
and each diagonal block acts upon a pair of entries of [ V] which are not both
zero. So, for example, when [1= _n_ = 3], the matrix can be
<math> \begin{pmatrix}
0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}.</math> For the vector
[(1, 0, 1/ [2] )], the resulting simplex has vertices
<math>
\begin{pmatrix} 1 \\ 0 \\ 1/\surd2 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\
-1/\surd2 \end{pmatrix}, \begin{pmatrix} -1 \\ 0 \\ 1/\surd2 \end{pmatrix},
\begin{pmatrix} 0 \\ -1 \\ -1/\surd2 \end{pmatrix}, </math> each of which has
distance 2 from the others.
** Geometric properties
*** Volume
The volume of an [n]-simplex in [n]-dimensional space with vertices [(_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 [_n_ × _n_] determinant is a vector that points from
vertex [_v_ [0]] to another vertex [_v_ [_k_]].[8] 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 [n]-simplex's vertices are in a Euclidean
space with more than [n] dimensions, e.g., a triangle in
<math>\mathbf{R}^3</math>.
A more symmetric way to compute the volume of an [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.[9]
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>(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 [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 [_n_ !] [n]-paths and <math>v_n</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.[10] In particular, the volume of such a simplex
is
<math> \frac{\operatorname{Vol}(P)}{n!} = \frac 1 {n!}. </math>
If [P] is a general parallelotope, the same assertions hold except that it is
no longer true, in dimension > 2, that the simplexes need to be pairwise
congruent; yet their volumes remain equal, because the [n]-parallelotope is
the image of the unit [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
[n]-path is:
<math> \frac{\operatorname{Vol}(P)}{n!} = \frac{\det(e_1, \ldots, e_n)}{n!}. </math>
Conversely, given an [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
[n]-simplex (i.e. between the origin and the simplex in [ R <sup> _n_ +1
</sup>]) is
<math> {1 \over (n+1)!} </math>
The volume of a regular [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 [_x_ <sup> _n_ +1
</sup>], 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>x=1/\sqrt{2}</math> (where the [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 [(_n_ − 1)]-dimensional faces of a regular [n]-dimensional simplex are
themselves regular [(_n_ − 1)]-dimensional simplices, and they have the same
dihedral angle of [cos <sup> −1 </sup> (1/ _n_ )].[11][12]
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 faces
are pairwise orthogonal. Such simplices are generalizations of right
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> \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.[13]
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 [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
[Categorical distribution]
In probability theory, the points of the standard [n]-simplex in [(_n_ +
1)]-space form the space of possible probability distributions on a finite set
consisting of [_n_ + 1] possible outcomes. The correspondence is as follows:
For each distribution described as an ordered [(_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 [k]th vertex of the simplex is assigned to have the [k]th probability
of the [(_n_ + 1)]-tuple as its barycentric coefficient. This correspondence
is an affine homeomorphism.
*** Aitchison geometry
[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 stella octangula .
- Two 5-cells form a compound of two 5-cells in four dimensions.
** Algebraic topology
<!--'Singular n-simplex' redirects here--> In algebraic topology, simplices
are used as building blocks to construct an interesting class of topological
spaces called simplicial complexes. These spaces are built from simplices
glued together in a combinatorial fashion. Simplicial complexes are used to
define a certain kind of homology called simplicial homology.
A finite set of [k]-simplexes embedded in an open subset of [ R <sup> _n_
</sup>] 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> \sigma= [ v_0,v_1,v_2,\ldots,v_n ] </math>
with the <math>v_j</math> denoting the vertices, then the boundary
<math>\partial\sigma</math> of [σ] is the chain
<math> \partial\sigma = \sum_{j=0}^n (-1)^j [ v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n ] . </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> \partial^2\sigma = \partial \left( \sum_{j=0}^n (-1)^j [ v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n ] \right) = 0. </math>
Likewise, the boundary of the boundary of a chain is zero: <math> \partial ^2
\rho =0 </math>.
More generally, a simplex (and a chain) can be embedded into a manifold by
means of smooth, differentiable map <math>f:\mathbf{R}^n \to M</math>. In
this case, both the summation convention for denoting the set, and the
boundary operation commute with the embedding. That is,
<math> f \left(\sum\nolimits_i a_i \sigma_i \right) = \sum\nolimits_i a_i f(\sigma_i) </math>
where the <math>a_i</math> are the integers denoting orientation and
multiplicity. For the boundary operator <math>\partial</math>, one has:
<math> \partial f(\rho) = f (\partial \rho) </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>f: \sigma \to X</math> to a topological space [X] is
frequently referred to as a SINGULAR [N] -SIMPLEX<!--boldface per WP:R#PLA-->.
(A map is generally called "singular" if it fails to have some desirable
property such as continuity and, in this case, the term is meant to reflect to
the fact that the continuous map need not be an embedding.)[14]
** Algebraic geometry
Since classical algebraic geometry allows one 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 display=block>\Delta^n := \left\{x \in
\mathbb{A}^{n+1} ~\Bigg|~ \sum_{i=1}^{n+1} x_i = 1\right\},</math> which
equals the scheme-theoretic description <math>\Delta_n(R) =
\operatorname{Spec}(R[\Delta^n])</math> with <math display=block>R[\Delta^n]
:= R[x_1,\ldots,x_{n+1}]\left/\left(1-\sum x_i \right)\right.</math> the ring
of regular functions on the algebraic [n]-simplex (for any ring
<math>R</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>R[\Delta^n]</math> assemble into one
cosimplicial object <math>R[\Delta^\bullet]</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
- In statistics , simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot .
- In probability theory , a simplex space is often used to represent the space of probability distributions. The Dirichlet distribution , for instance, is defined on a simplex.
- 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 mixture s, 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 .
[15]
- In operations research , linear programming problems can be solved by the simplex algorithm of George Dantzig .
- In game theory , strategies can be represented as points within a simplex. This representation simplifies the analysis of mixed strategies.
- In geometric design and computer graphics , many methods first perform simplicial triangulation s of the domain and then fit interpolating polynomials to each simplex. [16]
- In chemistry , the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point , fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron , and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
- In some approaches to quantum gravity , such as Regge calculus and causal dynamical triangulation s, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifold s.
** See also
[colwidth=12em]
- 3-sphere
- Aitchison geometry
- Causal dynamical triangulation
- Complete graph
- Delaunay triangulation
- Distance geometry
- Geometric primitive
- Hill tetrahedron
- Hypersimplex
- List of regular polytopes
- Metcalfe's law
- Other regular [n] - polytope s
- Cross-polytope
- Hypercube
- Tesseract
- Polytope
- Schläfli orthoscheme
- Simplex algorithm – an optimization method with inequality constraints
- Simplicial complex
- Simplicial homology
- Simplicial set
- Spectrahedron
- Ternary plot
[colend]
** Notes
[30em]
** References
- [author-link=Walter Rudin] _(See chapter 10 for a simple review of topological properties.)_
- [author-link=Andrew S. Tanenbaum]
- [first=Luc]
- [last=Coxeter]
- pp. 120–121, §7.2. see illustration 7-2 <small> A </small>
- p. 296, Table I (iii): Regular Polytopes, three regular polytopes in [n] dimensions ( [_n_ ≥ 5] )
- [urlname=Simplex]
- [author-link=Stephen P. Boyd] As PDF (see https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf)
[Wiktionary]
[Dimension topics] [Polytopes]
Category:Multi-dimensional geometry Category:Polytopes Category:Topology