In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron, and a 4-dimensional simplex is a 5-cell.
Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points are affinely independent, which means that the vectors are linearly independent. Then, the simplex determined by them is the set of points
A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex is the -dimensional simplex whose vertices are the standard unit vectors in or, in other words,
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex.
The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets.
History The concept of a simplex was known to William Kingdon Clifford, who introduced the term "prime confine" for these shapes in 1866 while solving a problem in geometric probability. Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").
The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as , the other two being the cross-polytope family, labeled as , and the hypercubes, labeled as . A fourth family, the tessellation of -dimensional space by infinitely many hypercubes, he labeled as .
Elements
The convex hull of any nonempty subset of the points that define an -simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size (of the defining points) is an -simplex, called an -face of the -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 ()-faces are called the facets, and the sole -face is the whole -simplex itself. In general, the number of -faces is equal to the binomial coefficient . Consequently, the number of -faces of an -simplex may be found in column () of row () of Pascal's triangle. A simplex is a coface of a simplex if is a face of . Face and facet can have different meanings when describing types of simplices in a simplicial complex.
The extended f-vector for an -simplex can be computed by , like the coefficients of polynomial products. For example, a 7-simplex is (1,1)8 = (1,2,1)4 = (1,4,6,4,1)2 = (1,8,28,56,70,56,28,8,1).
The number of 1-faces (edges) of the -simplex is the -th triangle number, the number of 2-faces of the -simplex is the th tetrahedron number, the number of 3-faces of the -simplex is the th 5-cell number, and so on.
+ -Simplex elements NameSchläfliCoxeter 0-faces(vertices) 1-faces(edges) 2-faces(faces) 3-faces(cells) 4-faces 5-faces 6-faces 7-faces 8-faces 9-faces 10-faces Sum= 2n+1 − 1 Δ0 0-simplex(point) ( ) 1 1 Δ1 1-simplex(line segment){ } = ( ) ∨ ( ) = 2⋅( ) 2 1 3 Δ2 2-simplex(triangle){3} = 3⋅( ) 3 3 1 7 Δ3 3-simplex(tetrahedron){3,3} = 4⋅( ) 4 6 4 1 15 Δ4 4-simplex(5-cell){33} = 5⋅( ) 5 10 10 5 1 31 Δ5 5-simplex{34} = 6⋅( ) 6 15 20 15 6 1 63 Δ6 6-simplex{35} = 7⋅( ) 7 21 35 35 21 7 1 127 Δ7 7-simplex{36} = 8⋅( ) 8 28 56 70 56 28 8 1 255 Δ8 8-simplex{37} = 9⋅( ) 9 36 84 126 126 84 36 9 1 511 Δ9 9-simplex{38} = 10⋅( ) 10 45 120 210 252 210 120 45 10 1 1023 Δ10 10-simplex{39} = 11⋅( ) 11 55 165 330 462 462 330 165 55 11 1 2047
An -simplex is the polytope with the fewest vertices that requires 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 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 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 a tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point 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 -simplex can be constructed as a join (∨ operator) of an -simplex and a point, . An -simplex can be constructed as a join of an -simplex and an -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: . A general 2-simplex (scalene triangle) is the join of three points: . An isosceles triangle is the join of a 1-simplex and a point: . An equilateral triangle is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: . A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: . A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: or . A regular tetrahedron is or and so on.
In some conventions,Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics) the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if . This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
Symmetric graphs of regular simplices These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges. 12345678910
Standard simplex
The standard -simplex (or unit -simplex) is the subset of given by .
The simplex lies in the affine hyperplane obtained by removing the restriction in the above definition.
The vertices of the standard -simplex are the points , where
⋮ .
A standard simplex is an example of a 0/1-polytope, with all coordinates as 0 or 1. It can also be seen one facet of a regular -orthoplex.
There is a canonical map from the standard -simplex to an arbitrary -simplex with vertices (, ..., ) given by
The coefficients are called the barycentric coordinates of a point in the -simplex. Such a general simplex is often called an affine -simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine -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 vertices) onto any polytope with vertices, given by the same equation (modifying indexing):
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex:
A commonly used function from to the interior of the standard -simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.
Examples Δ0 is the point in . Δ1 is the line segment joining and in . Δ2 is the equilateral triangle with vertices , and in . Δ3 is the regular tetrahedron with vertices , , and in . Δ4 is the regular 5-cell with vertices , , , and in .
Increasing coordinates An alternative coordinate system is given by taking the indefinite sum:
This yields the alternative presentation by order, namely as nondecreasing -tuples between 0 and 1:
Geometrically, this is an -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 corresponds to the inequalities becoming strict (increasing sequences).
A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the -cube, meaning that the orbit of the ordered simplex under the ! elements of the symmetric group divides the -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 1, , , , ..., .
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.
Projection onto the standard simplex Especially in numerical applications of probability theory, a projection onto the standard simplex is of interest. Given , possibly with coordinates that are negative or in excess of 1, the closest point on the simplex has coordinates
where is chosen such that
can be easily calculated from sorting the coordinates of . 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. Also see Integer programming.
Corner of cube Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
This yields an -simplex as a corner of the -cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with facets.
Cartesian coordinates for a regular -dimensional simplex in Rn