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 [ edit ] 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 ∧ {\displaystyle \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 ∧ {\displaystyle \wedge } m n denote its n th component, itself a finite (m − 1) -simplex with the edge length n , with a notational equivalent △ n m − 1 {\displaystyle \vartriangle _{n}^{m-1}} .
n th component[ edit ] ∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n :
| x | n = ∑ | k | = n ( n k ) x k ; x ∈ R m , k ∈ N 0 m , n ∈ N 0 , m ∈ N {\displaystyle |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} } where | x | = ∑ i = 1 m x i , | k | = ∑ i = 1 m k i , x k = ∏ i = 1 m x i k i {\displaystyle \textstyle |x|=\sum _{i=1}^{m}{x_{i}},\ |k|=\sum _{i=1}^{m}{k_{i}},\ x^{k}=\prod _{i=1}^{m}{x_{i}^{k_{i}}}} .
Example for ⋀4 [ edit ] Pascal's 4-simplex (sequence A189225 in the OEIS ), sliced along the k 4 . 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 [ edit ] Pascal's 1-simplex [ edit ] ∧ {\displaystyle \wedge } 1 is not known by any special name.
n th component[ edit ] ∧ n 1 = △ n 0 {\displaystyle \wedge _{n}^{1}=\vartriangle _{n}^{0}} (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n :
( x 1 ) n = ∑ k 1 = n ( n k 1 ) x 1 k 1 ; k 1 , n ∈ N 0 {\displaystyle (x_{1})^{n}=\sum _{k_{1}=n}{n \choose k_{1}}x_{1}^{k_{1}};\ \ k_{1},n\in \mathbb {N} _{0}} Arrangement of △ n 0 {\displaystyle \vartriangle _{n}^{0}} [ edit ] ( n n ) {\displaystyle \textstyle {n \choose n}} which equals 1 for all n .
Pascal's 2-simplex [ edit ] ∧ 2 {\displaystyle \wedge ^{2}} is known as Pascal's triangle (sequence A007318 in the OEIS ).
n th component[ edit ] ∧ n 2 = △ n 1 {\displaystyle \wedge _{n}^{2}=\vartriangle _{n}^{1}} (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 = ∑ k 1 + k 2 = n ( n k 1 , k 2 ) x 1 k 1 x 2 k 2 ; k 1 , k 2 , n ∈ N 0 {\displaystyle (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}} Arrangement of △ n 1 {\displaystyle \vartriangle _{n}^{1}} [ edit ] ( n n , 0 ) , ( n n − 1 , 1 ) , ⋯ , ( n 1 , n − 1 ) , ( n 0 , n ) {\displaystyle \textstyle {n \choose n,0},{n \choose n-1,1},\cdots ,{n \choose 1,n-1},{n \choose 0,n}} Pascal's 3-simplex [ edit ] ∧ 3 {\displaystyle \wedge ^{3}} is known as Pascal's tetrahedron (sequence A046816 in the OEIS ).
n th component[ edit ] ∧ n 3 = △ n 2 {\displaystyle \wedge _{n}^{3}=\vartriangle _{n}^{2}} (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 = ∑ k 1 + k 2 + k 3 = n ( n 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 ∈ N 0 {\displaystyle (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}} Arrangement of △ n 2 {\displaystyle \vartriangle _{n}^{2}} [ edit ] ( n n , 0 , 0 ) , ( n n − 1 , 1 , 0 ) , ⋯ ⋯ , ( n 1 , n − 1 , 0 ) , ( n 0 , n , 0 ) ( n n − 1 , 0 , 1 ) , ( n n − 2 , 1 , 1 ) , ⋯ ⋯ , ( n 0 , n − 1 , 1 ) ⋮ ( n 1 , 0 , n − 1 ) , ( n 0 , 1 , n − 1 ) ( n 0 , 0 , n ) {\displaystyle {\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}}} Properties [ edit ] Inheritance of components [ edit ] ∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} is numerically equal to each (m − 1) -face (there is m + 1 of them) of △ n m = ∧ n m + 1 {\displaystyle \vartriangle _{n}^{m}=\wedge _{n}^{m+1}} , or:
∧ n m = △ n m − 1 ⊂ △ n m = ∧ n m + 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}\subset \ \vartriangle _{n}^{m}=\wedge _{n}^{m+1}} From this follows, that the whole ∧ m {\displaystyle \wedge ^{m}} is (m + 1) -times included in ∧ m + 1 {\displaystyle \wedge ^{m+1}} , or:
∧ m ⊂ ∧ m + 1 {\displaystyle \wedge ^{m}\subset \wedge ^{m+1}} Example [ edit ] ∧ 1 {\displaystyle \wedge ^{1}} ∧ 2 {\displaystyle \wedge ^{2}} ∧ 3 {\displaystyle \wedge ^{3}} ∧ 4 {\displaystyle \wedge ^{4}} ∧ 0 m {\displaystyle \wedge _{0}^{m}} 1 1 1 1 ∧ 1 m {\displaystyle \wedge _{1}^{m}} 1 1 1 1 1 1 1 1 1 1 ∧ 2 m {\displaystyle \wedge _{2}^{m}} 1 1 2 1 1 2 1 2 2 1 1 2 1 2 2 1 2 2 2 1 ∧ 3 m {\displaystyle \wedge _{3}^{m}} 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 [ edit ] Conversely, ∧ n m + 1 = △ n m {\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}} is (m + 1) -times bounded by △ n m − 1 = ∧ n m {\displaystyle \vartriangle _{n}^{m-1}=\wedge _{n}^{m}} , or:
∧ n m + 1 = △ n m ⊃ △ n m − 1 = ∧ n m {\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}\supset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}} 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:
∧ n i + 1 = △ n i ⊂ △ n m > i = ∧ n m > i + 1 {\displaystyle \wedge _{n}^{i+1}=\vartriangle _{n}^{i}\subset \vartriangle _{n}^{m>i}=\wedge _{n}^{m>i+1}} Example [ edit ] 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 = ∧ n 1 = △ n 0 ⊂ △ n m − 1 = ∧ n m {\displaystyle 1=\wedge _{n}^{1}=\vartriangle _{n}^{0}\subset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}} Number of coefficients [ edit ] 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:
( ( n − 1 ) + ( m − 1 ) ( m − 1 ) ) + ( n + ( m − 2 ) ( m − 2 ) ) = ( n + ( m − 1 ) ( m − 1 ) ) = ( ( m n ) ) , {\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),} (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 [ edit ] Number of coefficients of n th component ((m − 1) -simplex) of Pascal's m -simplex m-simplex n th 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 [ edit ] An n th component ((m − 1) -simplex) of Pascal's m -simplex has the (m !)-fold spatial symmetry.
Geometry [ edit ] Orthogonal axes k 1 , ..., k m in m -dimensional space, vertices of component at n on each axis, the tip at [0, ..., 0] for n = 0 .
Numeric construction [ edit ] Wrapped n th power of a big number gives instantly the n th component of a Pascal's simplex.
| b d p | n = ∑ | k | = n ( n k ) b d p ⋅ k ; b , d ∈ N , n ∈ N 0 , k , p ∈ N 0 m , p : p 1 = 0 , p i = ( n + 1 ) i − 2 {\displaystyle \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}} where b d p = ( b d p 1 , ⋯ , b d p m ) ∈ N m , p ⋅ k = ∑ i = 1 m p i k i ∈ N 0 {\displaystyle \textstyle b^{dp}=(b^{dp_{1}},\cdots ,b^{dp_{m}})\in \mathbb {N} ^{m},\ p\cdot k={\sum _{i=1}^{m}{p_{i}k_{i}}}\in \mathbb {N} _{0}} .