Tuesday, September 21

Properties of Pascals Pyramid

Pascal’s Pyramid, or Pascal’s tetrahedron is an interesting extension of the
ideas from Pascal’s triangle. In this article, I take some of the basic
properties of Pascal’s triangle, such as sums of rows and see if they can be
modified to apply to Pascal’s tetrahedron.

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.