Properties of Pascal's Pyramid

One famous property of Pascal's triangle is that the sums of the rows are the doubling numbers. Rather than looking at the sums of rows in Pascal's pyramid, we can see if we get any similar patterns when we look at the sums of layers. This has been done for layers 0 to 4 below:

Layer 0:

1

Total = 1

Layer 1:

1

1 1

Total = 1 + 1 + 1 = 3

Layer 2:

1

2 2

1 2 1

Total = 1 + 2 + 2 + 1 + 2 + 1 = 9

Layer 3:

1

3 3

3 6 3

1 3 3 1

Total = 1 + 3 + 3 + 3 + 6 + 3 + 1 + 3 + 3 + 1 = 27

Layer 4:

1

4 4

6 12 6

4 12 12 4

1 4 6 4 1

Total = 1 + 4 + 4 + 6 + 12 + 6 + 4 + 12 + 12 + 4 + 1 + 4 + 6 + 4 + 1 = 81

The sums of the layers triple each time, producing a formula for the sum of the nth layer of 3^n. In fact, this property can tell us something else about Pascal's pyramid. Things can easily get very complicated with this, so I'm not going to try too explain this too much. If we look at the average of the numbers in each row of Pascal's triangle, we get the following results for the first few rows:

1, 1, 1.33, 2, 3.2, 5.33, 9.14...

Now, if we write down what you have to multiply each term by to get to next, you get

1, 1.33, 1.5, 1.6, 1.67, 1.71, 1.75...

If you kept on going, you would get a value closer and closer to 2, so you would get the average of the numbers in the rows eventually doubling each time.

If you try this with the average of the layers in Pascal's tetrahedron, you should find you get a sequence which gets closer and closer to tripling each time. This explains why the numbers get large so much faster in Pascal's tetrahedron.

We can also look at the symmetry of Pascal's tetrahedron. If you arrange each layer as an equilateral triangle, it has rotational symmetry of order 3, and reflective symmetry from each of its corners to the midpoint of the opposite side. This may sound complicated, but let's think about Pascal's triangle for a moment. in each row, every number appears twice unless it is in the very centre of the row. This is due to the symmetry through the centre of the triangle. Pascal's tetrahedron, however, due to slightly more complicated symmetry, has every number repeated three times is each layer, with the exception of a number which is sometimes found in the very centre of the triangular layer, such as the 6 in layer 3.

It is clear, therefore, that many of the properties of Pascal's triangle apply in some way to Pascal's pyramid as well, but it is interesting to think about how these patterns have evolved to suit Pascal's tetrahedron, and the reasons for these changes.