In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.

Generic Pascal's m-simplex

Let m (m > 0) be a number of terms of a polynomial and n (n ≥ 0) be a power the polynomial is raised to. Let \wedgem denote a Pascal's m-simplex. Each Pascal's m-simplex is a semi-infinite object, which consists of an infinite series of its components. Let \wedgem n denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent.

nth component

consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n: where.

Example for ⋀4

Pascal's 4-simplex, sliced along the k4. All points of the same color belong to the same nth component, from red (for n = 0) to blue (for n = 3).

Specific Pascal's simplices

Pascal's 1-simplex

\wedge1 is not known by any special name.

nth component

(a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

Arrangement of \vartriangle^0_n

which equals 1 for all n.

Pascal's 2-simplex

\wedge^2 is known as Pascal's triangle.

nth component

(a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

Arrangement of \vartriangle^1_n

Pascal's 3-simplex

\wedge^3 is known as Pascal's tetrahedron.

nth component

(a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

Arrangement of \vartriangle^2_n

Properties

Inheritance of components

is numerically equal to each (m − 1)-face (there is m + 1 of them) of, or: From this follows, that the whole \wedge^m is (m + 1)-times included in, or:

Example

For more terms in the above array refer to

Equality of sub-faces

Conversely, is (m + 1)-times bounded by, or: From this follows, that for given n, all i-faces are numerically equal in nth components of all Pascal's (m > i)-simplices, or:

Example

The 3rd component (2-simplex) of Pascal's 3-simplex is bounded by 3 equal 1-faces (lines). Each 1-face (line) is bounded by 2 equal 0-faces (vertices): 2-simplex 1-faces of 2-simplex 0-faces of 1-face 1 3 3 1 1 . . . . . . 1 1 3 3 1 1 . . . . . . 1 3 6 3 3 . . . . 3 . . . 3 3 3 . . 3 . . 1 1 1. Also, for all m and all n:

Number of coefficients

For the nth component ((m − 1)-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by: (where the latter is the multichoose notation). We can see this either as a sum of the number of coefficients of an (n − 1)th component ((m − 1)-simplex) of Pascal's m-simplex with the number of coefficients of an nth component ((m − 2)-simplex) of Pascal's (m − 1)-simplex, or by a number of all possible partitions of an nth power among m exponents.

Example

The terms of this table comprise a Pascal triangle in the format of a symmetric Pascal matrix.

Symmetry

An nth component ((m − 1)-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Geometry

Orthogonal axes k1, ..., km in m-dimensional space, vertices of component at n on each axis, the tip at [0, ..., 0] for n = 0.

Numeric construction

Wrapped nth power of a big number gives instantly the nth component of a Pascal's simplex. where.