Pascal's pyramid

In mathematics, Pascal's pyramid is a three dimensional generalization of Pascal's triangle. Just as Pascal's triangle gives coefficients for the terms of a binomial expansion, so Pascal's pyramid gives coefficients for a trinomial expansion.

Pascal's pyramid is also called Pascal's tetrahedron. A generalization to arbitrary dimension is sometimes called a Pascal's simplex.

The pyramid is constructed layer by layer, starting with the apex and working downward. Each edge is the appropriate row of Pascals triangle and each internal entry is the sum of three above it. The first few layers are as follows:

1      1         1              1                   1                      1 

     1   1     2   2          3   3               4   4                 5    5

             1   2   1      3   6   3           6  12   6             10  20  10

                          1   3   3   1       4  12   12  4         10  30  30  10

                                            1   4   6   4   1      5  20  30  20  5

                                                                 1   5  10  10  5   1

Here is a diagram showing how level five arises from level four. The black dots stand for 12 (level four) and the red dots for sites on level five which are the sum of the adjacent numbers from level four.

While studying genetics, it's common to use the Pascal's pyramid to find out the proportion between different genotypes on the same crossing. This is done by checking the line that is equivalent to the number of phenotypes (genotypes + 1). That line will be the proportion.

Properties of Pascal's pyramid

Summing the numbers in each column of a layer of Pascal's pyramid gives the n^th power of 111 when a sufficiently high base/radix is chosen (i.e. without carrying over during multiplication), where n is the layer minus 1.

 1     |1|     | |1| |      | | |1| | |       | |  |  | 1|  |  | |
---   1| |1    |2| |2|      | |3| |3| |       | |  | 4|  | 4|  | |
 1    -----   1| |2| |1     |3| |6| |3|       | | 6|  |12|  | 6| |
      1 1 1   ---------    1| |3| |3| |1      |4|  |12|  |12|  |4|
              1 2 3 2 1    -------------     1| | 4|  | 6|  | 4| |1
                           1 3 6 7 6 3 1     ----------------------
                                             1 4 10 16 19 16 10 4 1
1110   1111      1112           1113             1114 = 151807041 **

** 1×10⁸ + 4×10⁷ + 10×10⁶ + 16×10⁵ + 19×10⁴ + 16×10³ + 10×10² + 4×10¹ + 1×10⁰ = 151807041

Another property of Pascal's pyramid is that the bottom row of each layer in the pyramid is a row in Pascal's triangle. Any of the layers of Pascal's pyramid can be constructed easily because each row is proportionate to the corresponding row of Pascal's triangle. (More generally each side of a layer is the corresponding row in Pascal's triangle).

See also

• Multinomial theorem
• Trinomial expansion
• Pascal's Triangle

External links

• Beyond Flatland: Geometry for the 21st Century. PART I: Pascal's Tetrahedron
• Pascal's Pyramid Or Pascal's Tetrahedron?
• Pascal's Triangle