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Pascal's Pyramid
This is a 3D version of Pascal's Triangle.

I discovered this structure before I found out it was Pascal’s pyramid.

The solution of a statement of the type (a + b + c)n can be drawn as a two dimensional triangular grid; the components of the solution being at the nodes of the grid.

Imagine the page lying horizontally and the following grid patterns placed vertically over one another with the first one at the top, in the form a three-sided pyramid. The pyramid is infinite in extent but here I only show the first five layers.

1
(a + b + c)0
(a + b + c)1

(a + b + c)2

(a + b + c)3

(a + b + c)4

Comparing the pyramid to Pascal's Triangle

As with Pascal's triangle just display the numbers but in this case, layer by layer.
1 1

1          1		 1

2         2

1         2         1		 1

3         3

3         6         3

1         3         3         1		 1

4         4

6        12         6

4        12       12        4

1         4         6         4         1

1

5         5

10      20      10

10      30      30      10

5        20      30       20       5

1         5      10       10       5        1 		 1

6        6

15       30       15

20       60       60       20

15       60       90       60      15

6       30       60       60       30       6

1        6         15       20       15        6        1		 etc.

Some properties of the pyramid

  • The numbers at the edges of the pyramid are all ones.
  • Each of the faces of the pyramid (but not the base) are Pascal's triangles. That is, a number at a surface node (not a node at an edge, unless you assume that the pyramid is embedded in a 'sea' of zeros) is the sum of the two nearest face numbers in the layer above.
  • A number at an internal node is the sum of the three nearest numbers in the layer above.

For example, the fifth layer (red), over the sixth layer (green), over the seventh layer (blue).

1

1
6              6
1
5              5
15            30            15
4              4
10            20            10
20            60            60            20
6             12             6
10            30            30            10
15            60            90            60            15
4            12            12            4
5            20            30             20            5
6            30            60            60            30            6
1              4              6              4              1
1              5            10            10            5              1
1            6             15            20            15             6            1

The bigger picture

Pascal’s Triangle and Pascal’s Pyramid are the two simplest members in an infinite series of multi-dimensional structures. The following ones in the series are given by:
  • (a + b + c + d)n
  • (a + b + c + d + e)n
  • etc.
Mike Holden - Sep 2005
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