Pascal's simplex explained

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 be a number of terms of a polynomial and n 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 denote its nth component, itself a finite -simplex with the edge length n, with a notational equivalent

\vartriangle ^{m -1} _{n} .

nth component

\wedge ^{m} _{n} = \vartriangle ^{m -1} _{n} 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\inR} ^{m, k\inN} ^{m} ^{0 , n\in N} _{0, m\inN} where style|x|=\sum ^{m{x} _{i}, |k| = \sum} ^{k _{i}} ^{i} }.

Example for ⋀⁴

Pascal's 4-simplex, sliced along the k₄. All points of the same color belong to the same nth component, from red (for) to blue (for).

Specific Pascal's simplices

Pascal's 1-simplex

¹ is not known by any special name.

nth 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 ^{n} _{1 )} = \sum _{k _{1=n}} {n \choose k _{1}} x ^{k _{1}} _{1} ; k _{1,} n \in N _{0}

Arrangement of

\vartriangle ^{0} _{n} style {n \choose n} which equals 1 for all n.

Pascal's 2-simplex

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

nth 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 ^{n} _{2 )} = \sum _{k _{1+k} _{2=n}} {n \choose k _{1,} k _{2}} x ^{k _{1}} _{1} x ^{k _{2}} _{2} ; k _{1,} k _{2,} n \in N _{0}

Arrangement of

\vartriangle ^{1} _{n} style {n \choose n, 0 }, {n \choose n - 1 , 1 }, … , {n \choose 1 , n - 1 }, {n \choose 0 , n}

Pascal's 3-simplex

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

nth 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 ^{n} _{3 )} = \sum _{k _{1+k} _{2+k} _{3=n}} {n \choose k _{1,} k _{2,} k _{3}} x ^{k _{1}} _{1} x ^{k _{2}} _{2} x ^{k _{3}} _{3} ; k _{1,} k _{2,} k _{3,} n \in N _{0}

Arrangement of

\vartriangle ^{2} _{n} \begin{align} style {n \choose n, 0 , 0 } &, style {n \choose n - 1 , 1 , 0 }, … … , {n \choose 1 , n - 1 , 0 }, {n \choose 0 , n, 0 }\\ style {n \choose n - 1 , 0 , 1 } &, style {n \choose n - 2 , 1 , 1 }, … … , {n \choose 0 , n - 1 , 1 }\\ &\vdots\\ style {n \choose 1 , 0 , n - 1 } &, style {n \choose 0 , 1 , n - 1 }\\ style {n \choose 0 , 0 , n} \end{align}

Properties

Inheritance of components

\wedge ^{m} _{n} = \vartriangle ^{m -1} _{n} is numerically equal to each -face (there is of them) of \vartriangle ^{m} _{n} = \wedge ^{m +1} _{n} , or:\wedge ^{m} _{n} = \vartriangle ^{m -1} _{n} \subset \vartriangle ^{m} _{n} = \wedge ^{m +1} _{n}

From this follows, that the whole

\wedge ^{m} is -times included in \wedge ^{m+1} , or:\wedge ^{m} \subset \wedge ^{m+1}

Example

	\wedge ^{1}	\wedge ^{2}	\wedge ^{3}	\wedge ^{4}
\wedge ^{m} _{0}	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1
\wedge ^{m} _{1}	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 1 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 1 1 1
\wedge ^{m} _{2}	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 2 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 2 1 2 2 1	style="margin:0;padding:1ex;border:none;background:transparent;"> 1 2 1 2 2 1 2 2 2 1
\wedge ^{m} _{3}	style="margin:0;padding:1ex;border:none;background:transparent;"> 1	style="margin:0;padding:1ex;border:none;background:transparent;">1 3 3 1	style="margin:0;padding:1ex;border:none;background:transparent;">1 3 3 1 3 6 3 3 3 1	style="margin:0;padding:1ex;border:none;background:transparent;">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

Equality of sub-faces

Conversely,

\wedge ^{m +1} _{n} = \vartriangle ^{m} _{n} is (-times bounded by \vartriangle ^{m -1} _{n} = \wedge ^{m} _{n} , or:\wedge ^{m +1} _{n} = \vartriangle ^{m} _{n} \supset \vartriangle ^{m -1} _{n} = \wedge ^{m} _{n}

From this follows, that for given n, all i-faces are numerically equal in nth components of all Pascal's -simplices, or:

\wedge ^{i +1} _{n} = \vartriangle ^{i} _{n} \subset \vartriangle ^{m>i} _{n} = \wedge ^{m>i +1} _{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 ^{m -1} _{n} = \wedge ^{m} _{n}

Number of coefficients

For the nth component (-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:

{(n -1 ) + (m -1 ) \choose (m -1 )} + {n + (m - 2 ) \choose (m - 2 )} = {n + (m - 1 ) \choose (m - 1 )} = \left( \binom{m}{n} \right), (where the latter is the multichoose notation). We can see this either as a sum of the number of coefficients of an th component (-simplex) of Pascal's m-simplex with the number of coefficients of an nth component (-simplex) of Pascal's -simplex, or by a number of all possible partitions of an nth power among m exponents.

Example

Number of coefficients of nth component (-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 (-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Geometry

Orthogonal axes k₁, ..., k_m in m-dimensional space, vertices of component at n on each axis, the tip at for .

Numeric construction

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

\left|b ^{dp} \right| ^{n = \sum} _{|k| = n} {\binom{n}{k}b ^{dp ⋅} };\ \ b,d\in\mathbb,\ n\in\mathbb_0,\ k,p\in\mathbb_0^m,\ p:\ p_1=0, p_i=(n+1)^where style b ^{dp} = (b ^{dp _{1}} , … ,b ^{dp _{m}} )\in N ^{m, p ⋅} k = {\sum ^{m{p} _{i} k _{i}}\inN} _{0} .