My Pascal’s Triangle Shawl

I finished my Pascal’s Triangle Shawl!

I’m very happy with how it turned out!

In fact, I was disappointed by how the top edge curled — until I wore it, and it forms into a sort of collar! Perfect!

I already explained the math behind the shawl in great detail.

So now I’ll just say that this is a color-coded representation of Pascal’s Triangle, with a color for each prime factor, and each number represented in a diamond with its prime factorization shown.

In Pascal’s Triangle (at least when it’s shown with the point down, as above), each number is the sum of the two numbers beneath it, with 1 on all the ends. So 1 is white in my shawl.

The color scheme I used for the rest was:

2 is turquoise.
3 is yellow.
5 is red.
7 is purple.
11 is pink.
13 is light blue.

I took it up to the 15th row. After that, entries had more than 6 factors, so it wouldn’t be as easy to get them all in.

Take a moment to enjoy the flow. 🙂 Each time we get to a prime, every number in that row has that prime as a factor.

And the next row has that prime factor in all but the ends, and so it continues, forming an inverse triangle of that color. (This is because of the distributive law, as I explained in my earlier post.)

Looking at this shawl simply makes me happy. And I’m tremendously proud of it. I think it’s safe to say that this is the first Pascal’s Triangle Shawl ever knitted. 🙂

But it won’t be the last! As I began the shawl, I wasn’t sure it wasn’t a bit too garish with all the bright colors right next to each other. At least in the prime factorization blanket, I had rows of white in between the numbers. Though now that it’s finished, I completely love it.

Anyway, I decided to make a second one — this time using shades of pink and purple, with only subtle differences, going from light to dark. The first one will be easier to use for explaining the math, but I think the second one may be prettier.

And last night, I got another idea about how to make the second one different. Instead of having blocks of color for each factor, I’m planning to alternate rows. I think that will blend the colors as you look at the shawl — and I think it will be very beautiful! Stay tuned!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Pascal’s Triangle Shawl to Row 10

Hooray! I’ve knitted my Pascal’s Triangle Shawl all the way to the 10th row!

Now, it’s not finished — I’m going up to 15 — but I can’t resist explaining it already. I think it is SO COOL! And even more patterns are going to pop out as I continue.

My mathematical knitting began with my Prime Factorization Sweater, done in intarsia, with Tahki’s Cotton Classic yarn. It shows the prime factorization of all the numbers from 2 to 100, using a different color for each prime, with 1, the background color, in white.

Later when the internet discovered my sweater, I made a Café Press Prime Factorization T-shirt so anyone can have the color-coded prime factorization of the numbers from 2 to 100.

Now, the trouble with intarsia, is you have to carry all the colors you use in any given row along the back of the sweater. And there are about a million ends to sew in at the end. But a couple years ago, I got a hankering to do something like this again, and it occurred to me that if I used stripes, I could deal with one color at a time. I made a reversible Prime Factorization Scarf, where the thickness of the stripes tells you how many times a factor occurs. It also uses a different color for each prime. This time 1 is black, and there is a black stripe between each successive number. Within each number, there is a two-row stripe for each factor. This is done in Plymouth Encore yarn.

Then my brother, even more mathematically minded than me (if you can believe that!) was going to become a father. His daughter needed a prime factorization blanket! And it occurred to me that it would be far easier to knit the design in Intrelac, using rows of diamonds. I went back to the nice soft Cotton Classic yarn, and white as 1, to be bright for the baby. I used garter rows to show how many factors of each color.

The Prime Factorization Blanket turned out fantastic! But the horrible part was giving it away.

I got to thinking. Intrelac naturally falls into a triangle shape. I instantly thought of something mathematical in the shape of a triangle — Pascal’s Triangle! And I have a special fondness for Pascal’s Triangle, having won a Chalk Talk competition on the Binomial Theorem at a Math Field Day when I was a junior in high school. The numbers in Pascal’s Triangle are the Binomial Coefficients from the Binomial Theorem.

And — here’s where I started getting excited — I knew that there are some fascinating patterns in Pascal’s Triangle. Why not show the prime factorization of each number in the triangle? That would show some of the patterns.

So I began my Pascal’s Triangle Shawl. The first thing I noticed when sketching it out is very cool. Even though the numbers in the middle of the triangle get hugely big quite quickly, they never have any prime factors bigger than the number on the end of the row. So if I take the shawl to row 15, I will only need colors for 1, 2, 3, 5, 7, 11, and 13. To show the prime factorization this way (the same as the blanket), I’ll use 12 x 12 squares, using garter stitch rows to show the factors, with smooth stockinette stitch between factors.

The numbers in Pascal’s Triangle can be calculated two ways. The first way, each number is just the sum of the two numbers above it. Starting with 1.

So the 0th row is 1.

The 1st row is 1 1.

The 2nd row is 1 2 1. We get the 2 by adding the 1 and 1 above it.

The 3rd row is 1 3 3 1.

The 4th row is 1 4 6 4 1.

The 5th row is 1 5 10 10 5 1

The 6th row is 1 6 15 20 15 6 1.

And so on. In the blanket, you can figure out what number each color represents by looking on the edges.

Here it is again:

You can see that I’ve used white for 1. 2 is blue. 3 is yellow. 5 is red. 7 is purple.

You can’t see the garter stitch rows too clearly in that picture, so here’s a close-up of a section:

If you look at the numbers on the bottom edge, 5 is the solid red diamond. Then 6 is next to it, 3 x 2, yellow and blue. Then comes 7, purple. Then 8, which is 2 x 2 x 2, so it’s three sections of blue. Then going out of the picture will be 9 = 3 x 3, so two sections of yellow.

In the center of the shawl, the cool thing is that every diamond represents the sum of the two diamonds that touch its lower edges. See the red and yellow diamond? That would be 5 x 3 = 15. It is the sum of the two diamonds touching its lower edges, which are 10 = 5 x 2 (red and blue) and 5 (red).

Here’s another detailed view, but this time I’ve written in the numbers:

In that picture, see how each number is the sum of the two diamonds below it?

And see how the factorization works? 70, for example, is 7 x 5 x 2, so the colors are purple, red, and blue. 126 = 7 x 3 x 3 x 2, so the colors are purple, two sets of yellow, and blue.

Okay, there are two very cool patterns that I’ve already noticed from looking at the shawl.

First, whenever you’re on a prime row (with a prime on both ends), ALL of the numbers in that row will have the prime as a factor. See how every number in the 3rd row has some yellow? And every number in the 5th row has some red? And every number in the 7th row has some purple?

The reason for that involves the second way you can build Pascal’s Triangle. The rth number in the nth row is the Combination nCr, the number of ways of forming subsets of size r from a set of size n.

Okay, if I’ve just lost everyone, I’ll use examples. The 3rd number in the 5th row can be calculated as 5x4x3/3x2x1 (= 60/6 = 10). The 2nd number in the 7th row is 7×6/2×1 = 42/2 = 21. The 4th number in the 10th row is 10x9x8x7/4x3x2x1 = 10x3x7 = 210. (You always have r factors in the denominator, starting from r and going down 1 each. We call that r! or r factorial. On top, you also have r factors, but they start with n.)

If n is a prime number, all the numbers in that row of Pascal’s Triangle will have n as a factor, and there’s no way it will cancel out with anything in the denominator (except on the very ends when you have 1).

But all that you will notice in the shawl is the color popping up, and you don’t even have to know why. In fact, I planned the shawl by figuring out the sums, and I’d forgotten about the combinations. So I was delighted when I saw that prime factors consistently show up in all prime rows. And then I remembered why.

The second beautiful pattern is related to the sums. The shawl nicely shows the distributive law. If two diamonds next to each other have a factor the same, the diamond above them which they both touch will have the same factor. That’s because ca + cb = c(a + b).

For example, 21 + 35 = 56
and 7×3 + 7×5 = 7(3 + 5) = 7×8

When you combine those two patterns, we’ve got some inverse triangles. Look at the big picture again:

Now focus on the diamonds with red in them. (Red is 5.)

On the row with 5 on the ends, 1 5 10 10 5 1, every number (except the 1s) has red in it. Well, by the distributive law, every number in the next row that touches two of these will have red in it. Those are the three middle numbers on the next row, 15 20 15. The next row will have red wherever it touches two of those, 35 and 35. And finally, we’ll have red in the diamond that touches those two, 70.

The same inverse triangle is going to happen with 7 and purple.

And today I started knitting the 11th row, using pink for 11. So fun! 🙂

Now, I must admit, I’m not particularly pleased with the overall look. The colors looked better in the blanket with rows of white between them. In the shawl, they’re all mashed together and it’s a little bit much with such bright colors. So when I finish this one, I’m planning to make a new one with more subtle differences. I found a wool yarn, Northampton from, that has enough slightly different shades of purple. So I’ll be using these colors.

(I still have one more color on order, because the first one I ordered didn’t really go with these.)

The second shawl won’t be quite as good for explaining Pascal’s Triangle, but I think it will be much prettier! I will have to discipline myself to finish the first one before I start it. (I can solve that, I suppose, by using the same needles.)

So there you have it! Pascal’s Triangle knitted into a shawl! I will definitely post again when I finish it!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

More about Mathematical Knitting

On Wednesday, I finished knitting a Prime Factorization Blanket for my new little niece.

In my post about the blanket, I explained how the colors show the prime factorization of each number from 2 to 99. But I didn’t talk about the patterns, and I want to say a little bit about that here.

In fact, the only reason the Prime Factorization Blanket isn’t quite as good as the Prime Factorization Sweater is that I can’t have rows of 8 on the back and rows of 2 and 3 on the sleeves.

And the Prime Factorization Scarf is good for getting the flow of the numbers.

However, I do think the patterns in the 10 by 10 grid are a little easier to see with the larger diamonds on the blanket. Here’s the complete blanket laid out:

Let’s start by looking at the diagonals. 11 is 1 bigger than the base of 10. So the color for 11, red, goes in a diagonal across the blanket from the bottom left to the top right.

9 is 1 less than the base of 10. 9 = 3 x 3, so every number with a factor of 9 has two sections of yellow, the color for 3. You can see the yellows going diagonally up the blanket from the bottom right to the top left.

Oh, and I nearly forgot the more obvious ones. Since 2 and 5 are multiples of 10, they line up in columns. Every second column has turquoise for 2, and every fifth has green for 5.

Once you’re used to focusing on one color, you can pick any color and watch how it distributes evenly around the blanket. Take 19 for example, dark pink. You can see it climb up the blanket from the bottom right to the top left on a steeper diagonal than the one for 9.

And if you look at the colors it’s paired with, first it matches with 2, then with 3, then with two sets of 2, then with 5.

Another fun pattern is that the columns are sets of numbers that are congruent mod 10. So if you add, subtract, multiply or divide any two numbers in the same columns, your result will be in the same column.

For example, 2 + 17 = 19. Well, 62 + 27 = 99. The numbers in the second equation are from the same columns as the numbers in the first equation.

And that’s only the beginning of the patterns you can find.

Now that I’ve mailed off the blanket, I’m consoling myself by getting excited about the Pascal’s Triangle Shawl I’m going to make.

Pascal’s Triangle is formed by starting with 1, then adding a row of numbers where each number below is the sum of the two numbers above it.

Here’s how it works:

1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

The cool thing? Even though you’ve got really big numbers, when you factor them, the only prime factors are ones that have already appeared in the triangle. So I only need colors for the prime factors 2, 3, 5, 7, 11, and 13.

I’m stopping at row 15, because on row 16, we’ve got numbers with more than 6 prime factors, and that’s the most I’m prepared to accommodate unless I make my diamonds bigger. But up to 15 is going to be lovely.

I chose colors yesterday and tried to order the same colors. I’m not sure the ones I have aren’t discontinued colors by now, but when my order arrives, I can get going. (In fact, I’ve already started with the colors I have, hoping I won’t have to take out too much.) Here’s the color scheme I chose, using Cotton Classic yarn by Tahki.

2 is on the bottom left, and then it goes around counterclockwise. So 2 will be pale pink, 3 will be rose, 5 will be red, 7 will be purple, 11 will be yellow, and 13 will be turquoise. And they are going to repeat in beautiful ways, just you wait!

I’ve already begun, though if it turns out that the pink I’ve ordered is a different shade from what I have, I’ll have to take out the square for 2 that I’ve begun. But I can’t stand waiting for the order!

You can see there the initial diamond for the first row: 1.

Second row has two diamonds for the second row: 1 1

Third row, I’ve knitted the first diamond for 1, and I’ve begun the next diamond, for 2.

When I get to the row with 4, I will start showing the prime factorization, so 4 will be listed as 2 x 2, with two sections of pink.

The way I’ll show the prime factorization will be exactly like the blanket, but the patterns will be very different, always with the diamond representing the sum of the two numbers on its lower edges.

And it will get cool on the top edge with numbers like 6435 = 3 x 3 x 5 x 11 x 13

I can’t wait to show pictures of the final result. I think it will be beautiful!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.