The first five layers of Pascal's 3-simplex (Pascal's pyramid). Each face (orange grid) is Pascal's 2-simplex (Pascal's triangle). Arrows show derivation of two example terms.
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 m 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 m n denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent .
Pascal's 4-simplex (sequence A189225 in the OEIS), 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).
is known as Pascal's triangle (sequence A007318 in the OEIS).
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
Pascal's 3-simplex
is known as Pascal's tetrahedron (sequence A046816 in the OEIS).
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
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 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
2 2
1
1 2 1 2 2 1
2 2 2
1
1
1 3 3 1
1 3 3 1
3 6 3
3 3
1
1 3 3 1 3 6 3 3 3 1
3 6 3 6 6 3
3 3 3
1
For more terms in the above array refer to (sequence A191358 in the OEIS)
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):
(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
Number of coefficients of nth 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
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.