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
are
affinely independent, which means
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
:
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
:
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 ''α
n'', 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 ''β
n'', 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 ''γ
n''. A fourth family, the
tessellation of ''n''-dimensional space by infinitely many hypercubes, he labeled as ''δ
n''.
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 t ...
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 ...
. 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
:
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 hyper ...
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
:
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
-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):
:
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:
A commonly used function from R
''n'' to the interior of the standard
-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
* Δ
0 is the point .
* Δ
1 is the line segment joining (1, 0) and (0, 1) in R
2.
* Δ
2 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 R
3.
* Δ
3 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 R
4.
* Δ
4 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 R
5.
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 ...
:
:
This yields the alternative presentation by ''order,'' namely as nondecreasing ''n''-tuples between 0 and 1:
:
Geometrically, this is an ''n''-dimensional subset of
(maximal dimension, codimension 0) rather than of
(codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing,
here correspond to successive coordinates being equal,
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
mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume
Alternatively, the volume can be computed by an iterated integral, whose successive integrands are
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
with possibly negative entries, the closest point
on the simplex has coordinates
:
where
is chosen such that
can be easily calculated from sorting
.
The sorting approach takes
complexity, which can be improved to
complexity via
median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the
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:
:
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
; and the fact that the angle subtended through the center of the simplex by any two vertices is
.
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 components ...
of R
''n'' by e
1 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:
:
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:
:
for
, and
:
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
.
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
:
where
, and
:
The side length of this simplex is
.
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
:
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
:
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
:
Applying this to the vector results in the simplex whose vertices are
:
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
:
and each diagonal block acts upon a pair of entries of which are not both zero. So, for example, when , the matrix can be
:
For the vector , the resulting simplex has vertices
:
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
:
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
is the origin.
The expression
:
employs a
Gram determinant
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
and works even when the ''n''-simplex's vertices are in a Euclidean space with more than ''n'' dimensions, e.g., a triangle in
.
A more symmetric way to compute the volume of an ''n''-simplex in
is
:
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
of
.
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 ...
of
, call a list of vertices
a ''n''-path if
:
(so there are ''n''! ''n''-paths and
does not depend on the permutation). The following assertions hold:
If ''P'' is the unit ''n''-hypercube, then the union of the ''n''-simplexes formed by the convex hull of each ''n''-path is ''P'', and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
:
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
to
. As previously, this implies that the volume of a simplex coming from a ''n''-path is:
:
Conversely, given an ''n''-simplex
of
, it can be supposed that the vectors
form a basis of
. Considering the parallelotope constructed from
and
, one sees that the previous formula is valid for every simplex.
Finally, the formula at the beginning of this section is obtained by observing that
:
From this formula, it follows immediately that the volume under a standard ''n''-simplex (i.e. between the origin and the simplex in R
''n''+1) is
:
The volume of a regular ''n''-simplex with unit side length is
:
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
(where the ''n''-simplex side length is 1), and normalizing by the length
of the increment,
, 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
, and the centers of its faces are coordinate permutations of
. Then, by symmetry, the vector pointing from
to
is perpendicular to the faces. So the vectors normal to the faces are permutations of
, 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.
:
where
are facets being pairwise orthogonal to each other but not orthogonal to
, which is the facet opposite the orthogonal corner.
For a 2-simplex the theorem is the
Pythagorean theorem
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
: