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'' (''m'' > 0) be a number of terms of a polynomial and ''n'' (''n'' ≥ 0) be a power the polynomial is raised to.

Let $\wedge^m$ 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 $\wedge^m_n$ denote its ''n''^th component, itself a finite (''m − 1'')- simplex with the edge length ''n'', with a notational equivalent $\vartriangle^_n$.

''n''^th component

$\wedge^m_n = \vartriangle^_n$ consists of the coefficients of multinomial expansion of a polynomial with ''m'' terms raised to the power of ''n'':
:$, x, ^n=\sum_;\ \ x\in\mathbb^m,\ k\in\mathbb^m_0,\ n\in\mathbb_0,\ m\in\mathbb$
where $\textstyle, x, =\sum_^m,\ , k, =\sum_^m,\ x^k=\prod_^m$.

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 (for ''n'' = 0) to blue (for ''n'' = 3).

Specific Pascal's simplices

Pascal's 1-simplex

$\wedge^1$ is not known by any special name.

''n''^th component

$\wedge^1_n = \vartriangle^0_n$ (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of ''n'':
:$(x_1)^n = \sum_ x_1^;\ \ k_1, n \in \mathbb_0$

= Arrangement of $\vartriangle^0_n$

=
:$\textstyle$
which equals 1 for all ''n''.

Pascal's 2-simplex

$\wedge^2$ is known as Pascal's triangle .

''n''^th component

$\wedge^2_n = \vartriangle^1_n$ (a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of ''n'':
:$(x_1 + x_2)^n = \sum_ x_1^ x_2^;\ \ k_1, k_2, n \in \mathbb_0$

= Arrangement of $\vartriangle^1_n$

=
:$\textstyle , , \cdots, ,$

Pascal's 3-simplex

$\wedge^3$ is known as Pascal's tetrahedron .

''n''^th component

$\wedge^3_n = \vartriangle^2_n$ (a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of ''n'':
:$(x_1 + x_2 + x_3)^n = \sum_ x_1^ x_2^ x_3^;\ \ k_1, k_2, k_3, n \in \mathbb_0$

= Arrangement of $\vartriangle^2_n$

=
:$\begin \textstyle &, \textstyle , \cdots\cdots, , \\ \textstyle &, \textstyle , \cdots\cdots, \\ &\vdots\\ \textstyle &, \textstyle \\ \textstyle \end$

Properties

Inheritance of components

$\wedge^m_n = \vartriangle^_n$ is numerically equal to each (''m'' − 1)-face (there is ''m'' + 1 of them) of $\vartriangle^m_n = \wedge^_n$, or:

:$\wedge^m_n = \vartriangle^_n \subset\ \vartriangle^m_n = \wedge^_n$

From this follows, that the whole $\wedge^m$ is (''m'' + 1)-times included in $\wedge^$, or:

:$\wedge^m \subset \wedge^$

Example

For more terms in the above array refer to

Equality of sub-faces

Conversely, $\wedge^_n = \vartriangle^m_n$ is (''m'' + 1)-times bounded by $\vartriangle^_n = \wedge^m_n$, or:

:$\wedge^_n = \vartriangle^m_n \supset \vartriangle^_n = \wedge^m_n$

From this follows, that for given ''n'', all ''i''-faces are numerically equal in ''n''^th components of all Pascal's (''m'' > ''i'')-simplices, or:

:$\wedge^_n = \vartriangle^i_n \subset \vartriangle^_n = \wedge^_n$

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'':

:$1 = \wedge^1_n = \vartriangle^0_n \subset \vartriangle^_n = \wedge^m_n$

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:

:$+ = = \left( \binom \right),$

(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 ''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

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

Symmetry

An ''n''^th 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 ,...,0for .

Numeric construction

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

:$\left, b^\^n=\sum_;\ \ b,d\in\mathbb,\ n\in\mathbb_0,\ k,p\in\mathbb_0^m,\ p:\ p_1=0, p_i=(n+1)^$

where $\textstyle b^ = (b^,\cdots,b^)\in\mathbb^m,\ p\cdot k=\in\mathbb{N}_0$.

Factorial and binomial topics
Triangles of numbers