Simplex

In geometry, a simplex 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 wrote about these shapes in 1886 but called them "prime confines".
Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra".
In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum and then with the same Latin adjective in the normal form simplex.
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 is an -simplex, called an -face of the -simplex. The 0-faces are called the vertices, 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 ⁸ = ⁴ = ² =.
The number of 1-faces 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.

	Name	Schläfli Coxeter	0- faces	1- faces	2- faces	3- faces	4- faces	5- faces	6- faces	7- faces	8- faces	9- faces	10- faces	Sum = 2ⁿ⁺¹ − 1
Δ⁰	0-simplex		1											1
Δ¹	1-simplex	= ∨ = 2⋅	2	1										3
Δ²	2-simplex	= 3⋅	3	3	1									7
Δ³	3-simplex	= 4⋅	4	6	4	1								15
Δ⁴	4-simplex	= 5⋅	5	10	10	5	1							31
Δ⁵	5-simplex	= 6⋅	6	15	20	15	6	1						63
Δ⁶	6-simplex	= 7⋅	7	21	35	35	21	7	1					127
Δ⁷	7-simplex	= 8⋅	8	28	56	70	56	28	8	1				255
Δ⁸	8-simplex	= 9⋅	9	36	84	126	126	84	36	9	1			511
Δ⁹	9-simplex	= 10⋅	10	45	120	210	252	210	120	45	10	1		1023
Δ¹⁰	10-simplex	= 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. 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. 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 tetrahedron ABCD, a shape in a 3-dimensional space. 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. 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 and adding a second point, which required the increase to 1-dimensional space.
More formally, an -simplex can be constructed as a join 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 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 . 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, the empty set is defined to be a -simplex. The definition of the simplex above still makes sense if. This convention is more common in applications to algebraic topology than to the study of polytopes.

Symmetric graphs of regular simplices

These Petrie polygons show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20

Standard simplex

The standard -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 onto any polytope with vertices, given by the same equation :
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.