In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Pascal's simplex is a generalisation of Pascal's triangle

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

 into arbitrary number of dimensions

Dimensions

Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POV-Ray to render some of the animations, and the films are release under a Creative Commons licence....

, based on the multinomial theorem

Multinomial theorem

In mathematics, the multinomial theorem says how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem to polynomials.-Theorem:...

.

Induction of Pascal's simplices

Each Pascal's m-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 is a semi-infinite

Semi-infinite

The term semi-infinite has several related meanings in various branches of pure and applied mathematics. It typically describes objects which are infinite or unbounded in some but not all possible ways.-In ordered structures and Euclidean spaces:...

 object, which consists of a semi-infinite series (n ≥ 0) of finite (m − 1)-simplices, where m is the number of terms of a polynomial and n is a power the polynomial is raised to.

Let be a semi-infinite

Semi-infinite

The term semi-infinite has several related meanings in various branches of pure and applied mathematics. It typically describes objects which are infinite or unbounded in some but not all possible ways.-In ordered structures and Euclidean spaces:...

 Pascal's m-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 and its n^th component , a finite (m − 1)-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

 with edge length n.

Pascal's 1-simplex

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


Arrangement of :


which equals 1 for all n.

Pascal's 2-simplex

is known as Pascal's triangle

Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...

.

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


Arrangement of :

Pascal's 3-simplex

is known as Pascal's tetrahedron.

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


Arrangement of :

Pascal's m-simplex

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


Example for m = 4

Inheritance of components

is numericaly equal to each (m − 1)-face (there is m + 1 of them) of , or:


From this follows, that the whole is (m + 1)-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 (m + 1)-times bounded by , or:


From this follows, that for given n, all i-faces are numericaly equal in n^th 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 n^th component ((m − 1)-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:


that is, either by 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 n^th component ((m − 2)-simplex) of Pascal's (m − 1)-simplex, or by a number of all possible partitions of an n^th power among m exponents.

Example

Number of coefficients of n^th component ((m − 1)-simplex) of Pascal's m-simplex

m-simplex	nth component	n = 0	n = 1	n = 2	n = 3	n = 4	n = 5
1-simplex	0-simplex	1	1	1	1	1	1
2-simplex	1-simplex	1	2	3	4	5	6
3-simplex	2-simplex	1	3	6	10	15	21
4-simplex	3-simplex	1	4	10	20	35	56
5-simplex	4-simplex	1	5	15	35	70	126
6-simplex	5-simplex	1	6	21	56	126	252

Interestingly, the terms of this table comprise a Pascal triangle in the format of a symmetric Pascal matrix

Pascal matrix

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, a lower-triangular matrix, or a symmetric matrix...

.

Symmetry

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

Geometry

(Orthogonal axes k_1 ... k_m in m-dimensional space, vertices of component at n on each axe, the tip at [0,...,0] for n=0.)

Numeric construction

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