Mathematics in Needlepoint – Creating Pascal’s Triangle

Updated May 24, 2019.
You may not think so, but many mathematicians like to play. Mathematical games often take the guise of interesting formulas for creating a series of numbers.

One of these is a delightful construction called Pascal’s Triangle. Yes, it’s math, but only addition and simple multiplication, so bear with me. It’s a triangle so as you go down from the summit, each row has more “cells” in it, as you go down, each new cell in a row is the sum of the two cells in the row above it.

As you can see from this picture, the tip of the triangle is 1. The second row has two cells, both 1’s. Things start to get interesting in the third row. It’s 1-2-1. The outsides are always 1 because 1 + nothing is still 1. But the middle cell has the two 1’s above, so it contains 1+1 or 2.

Mentally work your way down the rows and see how this works out.

Clever isn’t it?

William Mitchell translated this to needlepoint in the piece pictured here. You can see a larger picture on this page.

Let’s look at those same top rows and see what he did. 1 is always a white square — see them running down both sides? The center block in the third row is 2 and is a red block, so red=2.

In the next row there are two yellow blocks. Looking at our number triangle we see that yellow=3.

Things really get interesting in the next row, where we have numbers that are not prime numbers. First we have 3+1, which is 4. But the square is a red square inside a red square. Of course 4 is also 2×2, so the red in red is 2×2!

The middle square in this row is 6. It’s not just 3+3, it’s also 2×3. What would we expect to see? Something with both red (2) and yellow (3) and that’s what’s there.

The next row has another prime number, 5, represented by a blue square. The other kind of square in this row is half red & half blue. It represents 10, or 2×5.

Got it?

I love to look at this design because the patterns get so complex. Every solid block is a prime number. The other blocks get more complicated as the factors get more complex. To use a simple case look at the middle block in the very next row. It is the sum of 10+10, or 20. The pattern tells use how to arrive at that number. It’s 2 times the product of 2×5. The more complicated patterns can indicate things like square or cubes of primes by the number of areas of a colors.

As you go down the pyramid, the patterns get more and more complex because the factors get more complex. This is what makes it so fascinating, both as a piece of art and as a piece of mathematics. You can love it just for the richness of pattern, but you can also love it because you can take any block in the design and follow the path up the pyramid to find the number it represents. Then you can work out the formula to get that number by figuring out the pattern.

Whenever I look at it I’m astonished by the work that went into planning and then stitching it. There are over 500 squares with each square requiring 120 stitches.

I’ll never do something so complex but I sure do wish I had. If you’ve created something based on a mathematical series, let me know in the comments.