Pascals Tetrahedron

An interesting extension of the ideas from Pascal's triangle can be found in Pascal's tetrahedron, a three-dimensional version of Pascal's triangle. In this article, I explain how Pascal's tetrahedron in formed and what links it has with Pascal's Triangle.

We are trying to create a triangular pyramid of numbers. Specifically, this should be a triangular based pyramid, not a square based pyramid like those in Egypt (there's nothing to stop you exploring a square based Pascal's pyramid, however, which is bound to have many interesting patterns, properties and links to the triangular version waiting to be discovered). At the very tip of the pyramid, we start with the number 1. Instead of looking down rows as in Pascal's triangle, we are interested in the layers of this pyramid, and each layer should be a triangle of numbers. Whilst in Pascal's triangle, each number is the sum of the two above, in Pascal's tetrahedron is the sum on numbers on the layer above.

It's easy to get confused at first when writing out the layers of Pascal's tetrahedron and thinking about what is supposed to add up to make what. Most people start off OK by writing down the first couple of layers like this:

Layer 0:

1

Layer 1:

1

1 1

Then, however, they want to add up all three numbers in layer 1 and put a 3 directly below the middle of them in layer 2. This is where they get confused as they can't make the numbers in layer 2 form a triangular shape.

What we actually need to do for layer 2 is take the sums of each of the three edges from layer 1 and also directly outwards, treating each number as a corner. Then, for layer 2 we get what is shown below:

1

2 2

1 2 1

Although in layer 2 we never added three numbers from the previous layer together, sometimes you have to. I find Pascal's pyramid very hard to visualise, so if you're finding this hard, then you're not alone! To avoid arranging and adding the numbers incorrectly, I have a couple of suggestions. Firstly, when writing out layers, centralise them rather than left justifying them. This makes is easier to see the symmetry in the layers and see triangles of numbers which you might have to add together.

Secondly, if you can, try to create some sort of model of Pascal's pyramid. This done most easily with cubes - if you have enough dice you can cover each one with paper and write your own numbers on them, and then stack them in a pyramid. This is, however, a little fiddly, so you may find it easier to draw equilateral triangles of dots of different sizes on separate laminates, tracing paper or thin tissue paper. If you use a different colour dot on each sheet, and place them on top of each other, you can see quite easily which dots from the top sheet are adjacent to any dot from the bottom sheet, telling what numbers you have to add together to calculate the numbers on the bottom sheet.

So you can check you've got the hang of it, I have listed the first the first few layers of Pascal's pyramid below:

Layer 0:

1

Layer 1:

1

1 1

Layer 2:

1

2 2

1 2 1

Layer 3:

1

3 3

3 6 3

1 3 3 1

Layer 4:

1

4 4

6 12 6

4 12 12 4

1 4 6 4 1

Layer 5:

1

5 5

10 20 10

10 30 30 10

5 20 30 20 5

1 5 10 10 5 1

Once you are confident at how Pascal's tetrahedron works, there is no end of fun to be had. The first and most obvious question is exactly how it links with Pascal's triangle. This can be done by thinking about how patterns from Pascal's triangle can be applied to Pascal's tetrahedron, and from there making comparisons between the two. Try to discover some new patterns and properties in the more complex world of Pascal's tetrahedron for yourself!

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