My Pascal’s Triangle Shawl

Mathematical, Prime Factorization 3

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.

The Prime Factorization Blanket!

Mathematical, Prime Factorization 2

Yes! My Masterpiece is finished!

What is this, you ask? This is a Prime Factorization Blanket!

With colors, it shows the prime factorization of all the integers from 1 to 99.

Here is the entire blanket, laid out flat:

Here’s how it works: Every prime number gets a color. The numbers start in the lower left corner.
I left a space for 0.
1 is the background color, white.
Then the next color is 2, a prime, so it gets its own color, blue.
3 is prime, and gets its own color, yellow.
4 is 2 x 2, so that square is two sections of blue. (You can tell on the blanket that there are two sections.)
5 is prime, and gets a new color, green.
6 = 2 x 3, so that square is part blue and part yellow. And so on.

I’ve got 0 through 9 on the first row, 10 through 19 in the next row, then 20 through 29, and so on through the top row, which is 90 through 99.

To show it more clearly, let’s look at each quadrant. Here’s the bottom left quadrant:

I put in the factors for each color. (After a few colors, I stopped putting in the “x” symbol for times.) I put a reference number on the left side so you can easily see which row. This set has 1 through 4, 10 through 14, 20 through 24, 30 through 34, and 40 through 44.

Now let’s look at the bottom right quadrant:

This picture shows 5 through 9, 15 through 19, 25 through 29, 35 through 39, and 45 through 49. For example, see if you can spot 48, which has a prime factorization of 2 x 2 x 2 x 2 x 3. Or look at 38, right below it, which equals 2 x 19.

By the way, this blanket is for my little niece, the daughter of my brother, who is, if it’s possible, even more of a math geek than me. On the 17th of December, my sister-in-law had an ultrasound, and we learned that the baby would be a girl, so I chose shades of pink for the next primes that came up, 17 and 19!

Now here’s the upper left quadrant:

This picture shows 50-54, 60-64, 70-74, 80-84, and 90-94. Can you find 62 = 2 x 31? Or 94 = 2 x 47? (I have to note that the colors are more distinct in person, and you can tell by the garter ridges how many sections there are of each color.)

And finally, the upper right quadrant:

And this, of course, covers 55-59, 65-69, 75-79, 85-89, and 95-99.

I’m so happy to finish it! The yarn is the same as what I used for my Prime Factorization Sweater, Cotton Classic. This yarn has enough colors (most important qualification), and it’s wonderfully soft — perfect for a baby blanket. I used a lot of leftover colors from the sweater, in fact.

The only really hard part? Giving it away! But I got the *idea* because my brother’s wife was having a baby, so this seems only fair to send it to the baby, as promised. Unfortunately, she lives on the other side of the country — so the one stipulation is they must take *lots* of pictures of her with it!

In fact, I thought of a way to console myself for giving away the blanket. My next project will be a Pascal’s Triangle Shawl!

I tested out, and the shape will work great!

I loved doing the entrelac squares for the blanket — it was much much easier than the intarsia I used on the Prime Factorization Sweater. And it will be easy-peasy to make a triangle instead of a square. I’ll use factors and do Pascal’s Triangle…. More on this to come, you can be sure!

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

Oh, and don’t forget! If you want your own prime factorization t-shirt or tote bag, you can find them at my Cafepress shop.

My Prime Factorization Scarf

Mathematical, Prime Factorization 2

I finished it! Last week, I finished sewing the ends in on my new Prime Factorization Scarf.

The scarf is similar to my Prime Factorization Sweater, using a new color for each prime factor. For the scarf, though, instead of making a grid of squares representing each number, I used two-row stripes for each factor. I separated each number with two rows of black, which represented the number 1 (since 1 times anything doesn’t change the value.)

I like the way the scarf gives the flow of the numbers. You can look closely at the blue color for 2 and watch it repeat. Then notice how the pink color for 3 repeats a little more slowly. And 5 a little more slowly than that. The scarf goes all the way up to 50.

Here are some sections up close. First, this picture shows 1 through 21:

2 is blue.
3 is pink.
4 = 2 x 2, so it’s two stripes of blue.
5 is yellow.
6 = 2 x 3, so it’s a stripe of blue and a stripe of pink.
7 is purple.
8 = 2 x 2 x 2, so it’s three stripes of blue.
9 = 3 x 3, so it’s two stripes of pink.
10 = 2 x 5, so it’s a stripe of blue and a stripe of yellow.
11 gets a new color, green.
12 = 2 x 2 x 3, so it’s two stripes of blue and a stripe of pink.
And so on….

Here is a picture showing 17 (light pink) through 35:

And finally, 33 to 50:

My earlier posts explained why I chose the pattern I did. I wanted the scarf to be reversible, but it’s not quite as easy to read as plain garter stitch stripes.

What’s next? A cuff-to-cuff cardigan! Only, I want to go higher than 50, so I decided to combine factors in one stripe — unless you have perfect powers of a number. Here’s a preview. I’m working on 33 now. (You can see that since 32 = 2^5, it’s 5 rows of blue.) It’s going to be flamboyantly bright, but I plan to wear my primes with pride!

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

Ready to Start My Prime Factorization Scarf!

Mathematical, Prime Factorization 1

My yarn arrived tonight! 26 shades of Plymouth Encore yarn (on sale at yarn.com), so I can make a Prime Factorization Scarf that goes all the way up to 100!

Now, a lot of the shades ended up looking more alike than I hoped they would. But I can always hold those toward the end where they only turn up once. I also didn’t realize what large skeins I was getting — I will need to make a sweater after this, because I’m going to have all kinds of leftover yarn. But I can change the color scheme to keep it interesting.

My mission first: Decide which colors will be most dominant. I’m planning on black for 1 this time, but I’m going to swatch out some different combinations for 2, 3, 5, and 7, to decide how I like it. I was planning on red for 2, but it’s so bright — I might not want that much red in the scarf. And I really like the turquoise blue that came. So we shall see… I’ll make some small swatches before I try the actual scarf.

If anyone wants to play along and make a scarf with me, let me know! It might be a lot smarter to make this as a leftover-yarn project and use up old yarn, instead of buying all the same yarn. I wish I’d thought of that! Anyway, I will think in terms of using the yarn for a cardigan later. For now, I’m looking forward to playing with some swatches!

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