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Pascal's simplex

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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

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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 .

nth component

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consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n:

where .

Example for ⋀4

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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).

First four components of Pascal's 4-simplex.

Specific Pascal's simplices

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Pascal's 1-simplex

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1 is not known by any special name.

First four components of Pascal's line.

nth component

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(a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

Arrangement of
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which equals 1 for all n.

Pascal's 2-simplex

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is known as Pascal's triangle (sequence A007318 in the OEIS).

First four components of Pascal's triangle.

nth component

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(a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

Arrangement of
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Pascal's 3-simplex

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is known as Pascal's tetrahedron (sequence A046816 in the OEIS).

First four components of Pascal's tetrahedron.

nth component

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(a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

Arrangement of
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Properties

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Inheritance of components

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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

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 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

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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

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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

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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

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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

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An nth component ((m − 1)-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Geometry

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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

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

where .