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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 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 (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 []
Pascal's 4-simplex (sequence A189225 in the OEIS), sliced along the k4. All points of the same color belong to the same n-th component, from red (for n = 0) to blue (for n = 3).
Specific Pascal's simplices[]
Pascal's 1-simplex[]
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 []
which equals 1 for all n.
Pascal's 2-simplex[]
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 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 (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):
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[]
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 in m-dimensional space, vertices of component at n on each axis, the tip at [0,...,0] for .
Numeric construction[]
Wrapped n-th power of a big number gives instantly the n-th component of a Pascal's simplex.
where .