HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a simplex (plural: simplexes or simplices) is a generalization of the notion of a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
or
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
to arbitrary
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
. The simplex is so-named because it represents the simplest possible
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
in any given dimension. For example, * a 0-dimensional simplex is a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
, * a 1-dimensional simplex is a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
, * a 2-dimensional simplex is a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
, * a 3-dimensional simplex is a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, and * a 4-dimensional simplex is a
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
. Specifically, a ''k''-simplex is a ''k''-dimensional
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
which is the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the
barycentric coordinate system In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The baryc ...
. A regular simplex is a simplex that is also a
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
. 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 is the ''k - 1'' dimensional simplex whose vertices are the ''k'' standard
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
, or :\left\. In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, it is common to "glue together" simplices to form a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
. The associated combinatorial structure is called an
abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
, in which context the word "simplex" simply means any
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...
of vertices.


History

The concept of a simplex was known to
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
, who wrote about these shapes in 1886 but called them "prime confines".
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, writing about
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
in 1900, called them "generalized tetrahedra". In 1902
Pieter Hendrik Schoute Pieter Hendrik Schoute (21 January 1846, Wormerveer – 18 April 1913, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. He started his career as a civil engineer, but became a professor of ...
described the concept first with the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
superlative ''simplicissimum'' ("simplest") and then with the same Latin adjective in the normal form ''simplex'' ("simple"). The regular simplex family is the first of three
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
families, labeled by
Donald Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
as ''αn'', the other two being the
cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
family, labeled as ''βn'', and the
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
s, labeled as ''γn''. A fourth family, the tessellation of ''n''-dimensional space by infinitely many hypercubes, he labeled as ''δn''.


Elements

The
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of any
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
\tbinom. 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 In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
. 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 In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
; see simplical complex for more detail. The number of 1-faces (edges) of the ''n''-simplex is the ''n''-th
triangle number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
, the number of 2-faces of the ''n''-simplex is the (''n'' − 1)th
tetrahedron number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
, the number of 3-faces of the ''n''-simplex is the (''n'' − 2)th 5-cell number, and so on. In layman's terms, an ''n''-simplex is a simple shape (a polygon) 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: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
is the join of a 1-simplex and a point:  ∨ ( ). An
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
is 3 ⋅ ( ) or . 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 ∨( ). A
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
is 4 ⋅ ( ) or and so on. 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 In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
) than to the study of polytopes.


Symmetric graphs of regular simplices

These
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
s (skew orthogonal projections) show all the vertices of the regular simplex on a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, and all vertex pairs connected by edges.


The standard simplex

The standard ''n''-simplex (or unit ''n''-simplex) is the subset of R''n''+1 given by : \Delta^n = \left\ The simplex Δ''n'' lies in the
affine hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
obtained by removing the restriction ''t''''i'' ≥ 0 in the above definition. The ''n'' + 1 vertices of the standard ''n''-simplex are the points ''e''''i'' ∈ R''n''+1, where :''e''0 = (1, 0, 0, ..., 0), :''e''1 = (0, 1, 0, ..., 0), : ⋮ :''e''''n'' = (0, 0, 0, ..., 1). There is a canonical map from the standard ''n''-simplex to an arbitrary ''n''-simplex with vertices (''v''0, ..., ''v''''n'') given by :(t_0,\ldots,t_n) \mapsto \sum_^n t_i v_i The coefficients ''t''''i'' are called the
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
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 In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
. 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 In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
with ''n'' vertices, given by the same equation (modifying indexing): :(t_1,\ldots,t_n) \mapsto \sum_^n t_i v_i These are known as
generalized barycentric coordinates In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The baryc ...
, and express every polytope as the ''image'' of a simplex: \Delta^ \twoheadrightarrow P. A commonly used function from R''n'' to the interior of the standard (n-1)-simplex is the
softmax function The softmax function, also known as softargmax or normalized exponential function, converts a vector of real numbers into a probability distribution of possible outcomes. It is a generalization of the logistic function to multiple dimensions, a ...
, or normalized exponential function; this generalizes the
standard logistic function A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
.


Examples

* Δ0 is the point . * Δ1 is the line segment joining (1, 0) and (0, 1) in R2. * Δ2 is the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R3. * Δ3 is the
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R4. * Δ4 is the regular
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
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 R5.


Increasing coordinates

An alternative coordinate system is given by taking the
indefinite sum In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or \Delta^ , is the linear operator, inverse of the forward difference operator \Delta . It relates to the forward difference operator ...
: : \begin 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_ + t_ = t_0 + t_1 + \cdots + t_\\ s_ &= s_n + t_n = t_0 + t_1 + \cdots + t_n = 1 \end This yields the alternative presentation by ''order,'' namely as nondecreasing ''n''-tuples between 0 and 1: :\Delta_*^n = \left\. Geometrically, this is an ''n''-dimensional subset of \mathbb^n (maximal dimension, codimension 0) rather than of \mathbb^ (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_, while the
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
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 Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
for the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
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 Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
a
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
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\, where \Delta is chosen such that \sum_i\max\=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\. 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 In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are ...
, 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''n''

One way to write down a regular ''n''-simplex in R''n'' 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 \pi/3; and the fact that the angle subtended through the center of the simplex by any two vertices is \arccos(-1/n). It is also possible to directly write down a particular regular ''n''-simplex in R''n'' which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
of R''n'' by e1 through e''n''. Begin with the standard -simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular -simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form for some
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
''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 In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
for ''α''. Solving this equation shows that there are two choices for the additional vertex: :\frac \left(1 \pm \sqrt \right) \cdot (1, \dots, 1). 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: :\frac\mathbf_i - \frac\bigg(1 \pm \frac\bigg) \cdot (1, \dots, 1), for 1 \le i \le n, and :\pm\frac \cdot (1, \dots, 1). Note that there are two sets of vertices described here. One set uses + in each calculation. The other set uses - in each calculation. This simplex is inscribed in a hypersphere of radius \sqrt. A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are :\sqrt\cdot\mathbf_i - n^(\sqrt \pm 1) \cdot (1, \dots, 1), where 1 \le i \le n, and :\pm n^ \cdot (1, \dots, 1). The side length of this simplex is \sqrt. A highly symmetric way to construct a regular -simplex is to use a representation of the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
by
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
. This is an orthogonal matrix such that is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, but no lower power of is. Applying powers of this
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
to an appropriate vector will produce the vertices of a regular -simplex. To carry this out, first observe that for any orthogonal matrix , there is a choice of basis in which is a block diagonal matrix :Q = \operatorname(Q_1, Q_2, \dots, Q_k), where each is orthogonal and either or . In order for to have order , all of these matrices must have order dividing . Therefore each is either a matrix whose only entry is or, if is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, ; or it is a matrix of the form :\begin \cos \frac & -\sin \frac \\ \sin \frac & \cos \frac \end, where each is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
between zero and inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices form a basis for the non-trivial irreducible real representations of , and the vector being rotated is not stabilized by any of them. In practical terms, for
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
this means that every matrix is , there is an equality of sets :\ = \, and, for every , the entries of upon which acts are not both zero. For example, when , one possible matrix is :\begin \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. Applying this to the vector results in the simplex whose vertices are : \begin 1 \\ 0 \\ 1 \\ 0 \end, \begin \cos(2\pi/5) \\ \sin(2\pi/5) \\ \cos(4\pi/5) \\ \sin(4\pi/5) \end, \begin \cos(4\pi/5) \\ \sin(4\pi/5) \\ \cos(8\pi/5) \\ \sin(8\pi/5) \end, \begin \cos(6\pi/5) \\ \sin(6\pi/5) \\ \cos(2\pi/5) \\ \sin(2\pi/5) \end, \begin \cos(8\pi/5) \\ \sin(8\pi/5) \\ \cos(6\pi/5) \\ \sin(6\pi/5) \end, each of which has distance √5 from the others. When is odd, the condition means that exactly one of the diagonal blocks is , equal to , and acts upon a non-zero entry of ; while the remaining diagonal blocks, say , are , there is an equality of sets :\left\ = \left\, and each diagonal block acts upon a pair of entries of which are not both zero. So, for example, when , the matrix can be :\begin 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \\ \end. For the vector , the resulting simplex has vertices : \begin 1 \\ 0 \\ 1/\surd2 \end, \begin 0 \\ 1 \\ -1/\surd2 \end, \begin -1 \\ 0 \\ 1/\surd2 \end, \begin 0 \\ -1 \\ -1/\surd2 \end, each of which has distance 2 from the others.


Geometric properties


Volume

The
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of an ''n''-simplex in ''n''-dimensional space with vertices (''v''0, ..., ''v''''n'') is : \mathrm = \frac \left, \det \begin v_1-v_0 && v_2-v_0 && \cdots && v_n-v_0 \end\ where each column of the ''n'' × ''n''
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
that points from vertex to another vertex . This formula is particularly useful when v_0 is the origin. The expression : \mathrm = \frac \det\left \begin v_1^T-v_0^T \\ v_2^T-v_0^T \\ \vdots \\ v_n^T-v_0^T \end \begin v_1-v_0 & v_2-v_0 & \cdots & v_n-v_0 \end \right 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 \mathbb^3. A more symmetric way to compute the volume of an ''n''-simplex in \mathbb^n is : \mathrm = \left, \det \begin v_0 & v_1 & \cdots & v_n \\ 1 & 1 & \cdots & 1 \end\. Another common way of computing the volume of the simplex is via the
Cayley–Menger determinant In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its v ...
, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions. 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 \R^n. Given a
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
\sigma of \, call a list of vertices v_0,\ v_1, \ldots, v_n a ''n''-path if :v_1 = v_0 + e_,\ v_2 = v_1 + e_,\ldots, v_n = v_+e_ (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 : \frac = \frac 1 . 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
that sends the canonical basis of \R^n to e_1,\ldots, e_n. As previously, this implies that the volume of a simplex coming from a ''n''-path is: : \frac = \frac. Conversely, given an ''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_ 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 is obtained 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_). From this formula, it follows immediately that the volume under a standard ''n''-simplex (i.e. between the origin and the simplex in R''n''+1) is : The volume of a regular ''n''-simplex with unit side length is :\frac as can be seen by multiplying the previous formula by ''x''''n''+1, to get the volume under the ''n''-simplex as a function of its vertex distance ''x'' from the origin, differentiating with respect to ''x'', at x=1/\sqrt  (where the ''n''-simplex side length is 1), and normalizing by the length dx/\sqrt of the increment, (dx/(n+1),\ldots, dx/(n+1)), 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 A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
of cos−1(1/''n''). This can be seen by noting that the center of the standard simplex is \left(\frac, \dots, \frac\right), and the centers of its faces are coordinate permutations of \left(0, \frac, \dots, \frac\right). Then, by symmetry, the vector pointing from \left(\frac, \dots, \frac\right) to \left(0, \frac, \dots, \frac\right) is perpendicular to the faces. So the vectors normal to the faces are permutations of (-n, 1, \dots, 1), 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 The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an ''n''-dimensional version of the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
: 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_^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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
for triangles with a right angle and for a 3-simplex it is
de Gua's theorem __NOTOC__ In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of th ...
for a tetrahedron with an orthogonal corner.


Relation to the (''n'' + 1)-hypercube

The
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ea ...
of the face lattice of an ''n''-simplex is isomorphic to the graph of the (''n'' + 1)-
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
'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 In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
of the (''n'' + 1)-hypercube. It is also the
facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
of the (''n'' + 1)-
orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
.


Topology

Topologically In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, an ''n''-simplex is
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivale ...
to an ''n''-ball. Every ''n''-simplex is an ''n''-dimensional
manifold with corners In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
.


Probability

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 In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
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.


Compounds

Since all simplices are self-dual, they can form a series of compounds; * Two triangles form a
hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
. * Two tetrahedra form a
compound of two tetrahedra In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra. Stellated octahedron There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral s ...
or
stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted ...
. * Two 5-cells form a
compound of two 5-cells In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
in four dimensions.


Algebraic topology

In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, simplices are used as building blocks to construct an interesting class of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s called
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es. These spaces are built from simplices glued together in a
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
fashion. Simplicial complexes are used to define a certain kind of
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
called
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
. A finite set of ''k''-simplexes embedded in an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of R''n'' is called an affine ''k''-chain. The simplexes in a chain need not be unique; they may occur with
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using multi ...
. 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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
, 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 Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
of an ''n''-simplex is an affine (''n'' − 1)-chain. Thus, if we denote one positively oriented affine simplex as :\sigma= _0,v_1,v_2,\ldots,v_n/math> with the v_j denoting the vertices, then the boundary \partial\sigma of ''σ'' is the chain :\partial\sigma = \sum_^n (-1)^j _0,\ldots,v_,v_,\ldots,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 \left( \sum_^n (-1)^j _0,\ldots,v_,v_,\ldots,v_n\right) = 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
by means of smooth, differentiable map f\colon\R^n \to M. In this case, both the summation convention for denoting the set, and the boundary operation commute with the
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
. That is, :f \left(\sum\nolimits_i a_i \sigma_i \right) = \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 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
f: \sigma \to X to a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' is frequently referred to as a singular ''n''-simplex. (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.)


Algebraic geometry

Since classical
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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 \Delta^n := \left\, which equals the
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
-theoretic description \Delta_n(R) = \operatorname(R Delta^n with R Delta^n:= R _1,\ldots,x_left/\left(1-\sum x_i \right)\right. the ring of regular functions on the algebraic ''n''-simplex (for any
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
R). By using the same definitions as for the classical ''n''-simplex, the ''n''-simplices for different dimensions ''n'' assemble into one
simplicial object In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
, while the rings R Delta^n/math> assemble into one cosimplicial object R Delta^\bullet/math> (in the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of schemes resp. rings, since the face and degeneracy maps are all polynomial). The algebraic ''n''-simplices are used in higher
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
and in the definition of higher
Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...
s.


Applications

*In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, simplices are sample spaces of
compositional data In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, ...
and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a
ternary plot A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equi ...
. *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 In chemistry, a mixture is a material made up of two or more different chemical substances which are not chemically bonded. A mixture is the physical combination of two or more substances in which the identities are retained and are mixed in the ...
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 In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM ...
, and then a local maximum can be computed using a
nonlinear programming In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or sta ...
method, such as
sequential quadratic programming Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable. SQP me ...
. *In
operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
,
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
problems can be solved by the
simplex algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
of
George Dantzig George Bernard Dantzig (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics. Dantzig is known for his dev ...
. *In geometric design and
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, many methods first perform simplicial
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
s of the domain and then fit interpolating
polynomials In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
to each simplex. *In
chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, the hydrides of most elements in the
p-block A block of the periodic table is a set of elements unified by the atomic orbitals their valence electrons or vacancies lie in. The term appears to have been first used by Charles Janet. Each block is named after its characteristic orbital: s-blo ...
can resemble a simplex if one is to connect each atom.
Neon Neon is a chemical element with the symbol Ne and atomic number 10. It is a noble gas. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with about two-thirds the density of air. It was discovered (along with krypton ...
does not react with hydrogen and as such is a point,
fluorine Fluorine is a chemical element with the symbol F and atomic number 9. It is the lightest halogen and exists at standard conditions as a highly toxic, pale yellow diatomic gas. As the most electronegative reactive element, it is extremely reacti ...
bonds with one hydrogen atom and forms a line segment,
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
bonds with two hydrogen atoms in a bent fashion resembling a triangle,
nitrogen Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
reacts to form a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, and
carbon Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with o ...
forms a structure resembling a
Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the origina ...
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 The halogens () are a group in the periodic table consisting of five or six chemically related elements: fluorine (F), chlorine (Cl), bromine (Br), iodine (I), astatine (At), and tennessine (Ts). In the modern IUPAC nomenclature, this group is ...
atom. *In some approaches to
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
, such as
Regge calculus In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961. Available ( ...
and
causal dynamical triangulation Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent. This means that it does not ass ...
s, simplices are used as building blocks of discretizations of spacetime; that is, to build
simplicial manifold In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard us ...
s.


See also

*
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
*
Aitchison geometry In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex. Measurements involving probabilities, proportions, ...
*
Causal dynamical triangulation Causal dynamical triangulation (abbreviated as CDT) theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent. This means that it does not ass ...
*
Complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...
*
Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
*
Distance geometry Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based ''only'' on given values of the distances between pairs of points. More abstractly, it is the study of semimetric spaces and the isome ...
*
Geometric primitive In vector computer graphics, CAD systems, and geographic information systems, geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that ...
*
Hill tetrahedron In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M. J. M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube. Const ...
*
Hypersimplex In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones ...
*
List of regular polytopes This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ' ...
*
Metcalfe's law Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (''n''2). First formulated in this form by George Gilder in 1993, and attributed to Robert Metcalf ...
* Other regular ''n''-
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
s **
Cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
**
Hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
**
Tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...
*
Polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
*
Schläfli orthoscheme In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges (v_0v_1), (v_1v_2), \dots, (v_v ...
*
Simplex algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
—a method for solving optimization problems with inequalities. *
Simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
*
Simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
*
Simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
*
Spectrahedron In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality. Alternatively, the set of positive semidefinite matrices forms a convex cone in , and a spectrahedron is a shape that can be formed by inters ...
*
Ternary plot A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equi ...


Notes


References

* ''(See chapter 10 for a simple review of topological properties.)'' * * * ** pp. 120–121, §7.2. see illustration 7-2A ** p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ''n'' dimensions (''n'' ≥ 5) * * A
PDF
{{Polytopes Polytopes Topology Multi-dimensional geometry