Pascal’s triangle and Pascal’s tetrahedron
I’ve started by showing the first 4 layers of Pascal’s tetrahedron below:
Layer 0:
1
Layer 1:
1
eleven
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 layer is 1,3,3,1, row three of Pascal’s triangle, the final row of layer 4 is 4, and so on for all the layers listed above. In fact, if you write each of the layers shown above in a centered equilateral triangle, you’ll notice that each edge of each triangular layer is the corresponding row of that layer in Pascal’s triangle.
This pattern, in fact, always continues. Without going too deep into rigorous math proofs, if you think about it, you should be able to see that at the edges of the layers, you just add two numbers from the top layer, and it’s actually like Pascal’s triangle (where you add two numbers from the row above) repeated three times at various angles.
However, the links go even deeper than this (in more ways than one). It’s not just the edges of the pyramid, there are links inside the core of the tetraheron itself.
To understand this, we’re going to use layer 4 as an example, but this time, we’re not looking at an edge but at some of the rows that run through the middle of the layer. For example, is there anything interesting in the penultimate row, which goes 4,12,12,4? In fact, there is (otherwise I wouldn’t be asking). For the moment, let’s forget about the numbers themselves and think only about the relationship among them. This gives a ratio of 1:3:3:1, since the two middle numbers are three of the two outer numbers. This ratio becomes the third row of Pascal’s triangle. Is it just a coincidence?
Next, let’s investigate the second to 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 entire layer 4, divided into rows, with their proportions 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 surprisingly, each row in layer 4 is in the proportion of the row in Pascal’s triangle that has the same number of numbers! However, just when you thought it couldn’t get more exciting, check out why we have to multiply ratios to get the real 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 fourth row of Pascal’s triangle! Only now do we really see the extent of the links between these two number patterns. Each number in the pyramid is simply two numbers in Pascal’s triangle multiplied. In my opinion, not only is this a beautiful discovery that is an excellent demonstration of the interconnected nature of mathematics, but it makes what seemed to be the much more complex Pascal tetrahedron easy to work with. In fact, it’s these links that have helped mathematicians modify Pascal’s triangle formula to one that applies to Pascal’s tetrahedron and even to higher dimensions, so it’s certainly a very powerful discovery!
