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!