Pascal's simplex


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 be a number of terms of a polynomial and n be a power the polynomial is raised to.
Let 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 denote its nth component, itself a finite -simplex with the edge length n, with a notational equivalent.

''n''th component

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

Example for \wedge^4

Pascal's 4-simplex, sliced along the k4. All points of the same color belong to the same n-th component, from red to blue.

Specific Pascal's simplices

Pascal's 1-simplex

is not known by any special name.

''n''th component

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

is known as Pascal's triangle.

''n''th component

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

is known as Pascal's tetrahedron.

''n''th component

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 -face of, or:
From this follows, that the whole is -times included in, or:

Example

1 1 1 1
1 1 1 1 1 1 1 1
1 1
1 1 2 1 1 2 1 1 2 1 2 2 1
2 2 2 2 2
1 1
1 1 3 3 1 1 3 3 1 1 3 3 1 3 6 3 3 3 1
3 6 3 3 6 3 6 6 3
3 3 3 3 3
1 1
For more terms in the above array refer to

Equality of sub-faces

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

Example

The 3rd component of Pascal's 3-simplex is bounded by 3 equal 1-faces. Each 1-face is bounded by 2 equal 0-faces :
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 of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:
. We can see this either as a sum of the number of coefficients of an th component of Pascal's m-simplex with the number of coefficients of an nth component of Pascal's -simplex, or by a number of all possible partitions of an nth power among m exponents.

Example

m-simplexnth componentn = 0n = 1n = 2n = 3n = 4n = 5
1-simplex0-simplex111111
2-simplex1-simplex123456
3-simplex2-simplex136101521
4-simplex3-simplex1410203556
5-simplex4-simplex15153570126
6-simplex5-simplex162156126252

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

Symmetry

An nth component of Pascal's m-simplex has the -fold spatial symmetry.

Geometry

Orthogonal axes in m-dimensional space, vertices of component at n on each axe, the tip at for.

Numeric construction

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