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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:
![|x|^{n}=\sum _{|k|=n}{{\binom {n}{k}}x^{k}};\ \ x\in \mathbb {R} ^{m},\ k\in \mathbb {N} _{0}^{m},\ n\in \mathbb {N} _{0},\ m\in \mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9cbe15478b26643f9b5235d8c37cef84be8b8)
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:
![(x_{1})^{n}=\sum _{k_{1}=n}{n \choose k_{1}}x_{1}^{k_{1}};\ \ k_{1},n\in \mathbb {N} _{0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71d462868c6d1341b52dcdf0166f78d9e132bde2)
Arrangement of
[]
![\textstyle {n \choose n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92856e04872073589a5ee9aa36534a70cba97aed)
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:
![(x_{1}+x_{2})^{n}=\sum _{k_{1}+k_{2}=n}{n \choose k_{1},k_{2}}x_{1}^{k_{1}}x_{2}^{k_{2}};\ \ k_{1},k_{2},n\in \mathbb {N} _{0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b8a1f750bbb40a29de9175a5fd413388890d33)
Arrangement of
[]
![\textstyle {n \choose n,0},{n \choose n-1,1},\cdots ,{n \choose 1,n-1},{n \choose 0,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6784fa940a845b9c3d5594ea08b41bf8dbca8e1)
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:
![(x_{1}+x_{2}+x_{3})^{n}=\sum _{k_{1}+k_{2}+k_{3}=n}{n \choose k_{1},k_{2},k_{3}}x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}};\ \ k_{1},k_{2},k_{3},n\in \mathbb {N} _{0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f28e67c29b32102928b60d3d25219dcff0bee0c)
Arrangement of
[]
![{\begin{aligned}\textstyle {n \choose n,0,0}&,\textstyle {n \choose n-1,1,0},\cdots \cdots ,{n \choose 1,n-1,0},{n \choose 0,n,0}\\\textstyle {n \choose n-1,0,1}&,\textstyle {n \choose n-2,1,1},\cdots \cdots ,{n \choose 0,n-1,1}\\&\vdots \\\textstyle {n \choose 1,0,n-1}&,\textstyle {n \choose 0,1,n-1}\\\textstyle {n \choose 0,0,n}\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28c263f163da3effc306762e985b420744f5db00)
Properties[]
Inheritance of components[]
is numerically equal to each (m − 1)-face (there is m + 1 of them) of
, or:
![\wedge _{n}^{m}=\vartriangle _{n}^{m-1}\subset \ \vartriangle _{n}^{m}=\wedge _{n}^{m+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e81fcbb90534c071082ea57ab885a1ef4fe5ee16)
From this follows, that the whole
is (m + 1)-times included in
, or:
![\wedge ^{m}\subset \wedge ^{m+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/670c0044fa6e2667546b794b6bb1c8cbc1b0f019)
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:
![\wedge _{n}^{m+1}=\vartriangle _{n}^{m}\supset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/577cb9c71874c0ed55ca6152bd012daa35bea78f)
From this follows, that for given n, all i-faces are numerically equal in nth components of all Pascal's (m > i)-simplices, or:
![\wedge _{n}^{i+1}=\vartriangle _{n}^{i}\subset \vartriangle _{n}^{m>i}=\wedge _{n}^{m>i+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7acec649c468801c76e3c5e78b1456c9d2232487)
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:
![1=\wedge _{n}^{1}=\vartriangle _{n}^{0}\subset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/727ad282d57c02c964ed1889c26ccf2796dd5ffd)
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:
![{\displaystyle {(n-1)+(m-1) \choose (m-1)}+{n+(m-2) \choose (m-2)}={n+(m-1) \choose (m-1)}=\left({\binom {m}{n}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4ec8ceb6dd8573d57b6136cda3612ab8b7c765)
(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.
![\left|b^{dp}\right|^{n}=\sum _{|k|=n}{{\binom {n}{k}}b^{dp\cdot k}};\ \ b,d\in \mathbb {N} ,\ n\in \mathbb {N} _{0},\ k,p\in \mathbb {N} _{0}^{m},\ p:\ p_{1}=0,p_{i}=(n+1)^{i-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e12c20acddb11f1addae4408b68fce276e8dd8)
where
.