Pascal's Triangle and Pascal's Tetrahedron

I have begun by showing the first 4 layers of Pascal's tetrahedron 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

In layer 3, the final row of the layer is 1,3,3,1, row three of Pascal's triangle, the final row of layer 4 the 4th, and so on for all the layers listed above. In fact, if you write out each of the layers shown above in a centered equilateral triangle, you will notice that every edge of each triangular layer is that layer's corresponding row in Pascal's triangle.

This pattern does, in fact, always continue. Without delving too deeply into rigorous mathematical proofs, you should if you think about it be able to see that on the edges of layers, you only ever add together two numbers from the layer above, and so really it's just like Pascal's triangle (where you add two numbers from the row above) repeated three times at various angles.

However, the links run even deeper than this (in more ways than one). Its not just about the edges of the pyramid, there are links deep down inside the very core of the tetraheron.

To understand this, we are going to use layer 4 as an example, but this time, we will not look at an edge but as some of the rows running through the middle of the layer. For example, is there anything interesting about the second to last row, which goes 4,12,12,4? In fact there is (otherwise I wouldn't be asking). For the moment, let's just forget about the numbers themselves, and think only about the ratio between them. This gives a 1:3:3:1 ratio, as the middle two numbers are thrice the outside two numbers. This ratio happens to be the third row of Pascal's triangle. Is that just a coincidence?

Next, let's investigate the third last row in layer 4, which goes 6,12,6. This time, the ratio is 1:2:1, the second row. Something is definitely going on here. Below is the whole of layer 4, split into rows, with their ratios and where they can be found in Pascal's triangle:

Layer 4:

1

4 4 - ratio 1:1 (row 1)

6 12 6 - ratio 1:2:1 (row 2)

4 12 12 4 - ratio 1:3:3:1 (row 3)

1 4 6 4 1 - ratio 1:4:6:4:1 (row 4)

So, amazingly, every single row in layer 4 is in the ratio of the row in Pascal's triangle which has the same number of numbers in it! However, just when you thought it couldn't get any more exciting, look at what we have to multiply the ratios by to get the actual numbers in Pascal's tetrahedron again:

1 (1) x 1

4 4 - (1,1) x 4

6 12 6 - (1,2,1) x 6

4 12 12 4 - (1,3,3,1) x 4

1 4 6 4 1 - (1,4,6,4,1) x 1

They are the 4th row of Pascal's triangle! Only now do we truly see the extent of the links between these two patterns of numbers. Every single number in the pyramid is simply two numbers from Pascal's triangle multiplied together. Not only is this in my opinion a beautiful discovery which is an excellent demonstration of the interconnected nature of mathematics, it makes what seemed like the much more complex Pascal's tetrahedron easy to work with. In fact, it is these links that have helped mathematicians to modify the formula for Pascal's triangle to one which applies to Pascal's tetrahedron and even to suit higher dimensions, so it is certainly a very powerful discovery!

Pascal's Tetrahedron

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!

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.