Category: Mathematical

I’ve finished the second row of numbers (third row of rectangles) in my Prime Factorization Blanket!

The fun part was that my brother and his wife found out on the 17th of December that their baby is a girl. So, since I was coming up on the prime number 17, I chose to use pastel pink to represent 17. For good measure, I used a pretty rose pink to represent 19.

I only hope that having all that turquoise blue won’t make people think it’s a blanket for a boy, but I’m hoping it’s multicolored enough, it won’t give that idea.

I couldn’t manage to write in all the numbers on the picture, like I did after the first row, but in real life I assure you, you can tell when there are two factors of the same prime.

So here’s how you read the blanket:

The bottom row starts with a blank space for 0.

1 is the same as the background color, since 1 is a factor of every number.

2 is turquoise blue.

3 is yellow.

4 = 2 x 2, so two sections of turquoise.

5 is green.

6 = 2 x 3, so a section of turquoise and a section of yellow.

7 is purple.

8 = 2 x 2 x 2, so three sections of turquoise.

9 = 3 x 3, so two sections of yellow.

Then I did a row of white rectangles (diamonds). Second row of color:

11 is red.

12 = 2 x 2 x 3, so two sections of turquoise and one of yellow.

13 is brown.

14 = 2 x 7, so a section of turquoise and a section of purple.

15 = 3 x 5, so a section of yellow and a section of green.

16 = 2 x 2 x 2 x 2, so four sections of turquoise.

17 is Pink!

18 = 2 x 3 x 3, so a section of turquoise and two sections of yellow.

19 is rose pink.

Next I’ll do a row of white rectangles, then start the next row with 20. The primes in that row will be 23 and 29, so I’ll have to bring in two new colors.

The color sections will show up better after I’ve knitted the white rectangles, but I was impatient to show what I’ve done!

I’m very pleased with how this is turning out. I may have to make myself a Pascal’s Triangle Shawl when I’m done….

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

Tweet

So, Tuesday I posted my plan for making a Prime Factorization Blanket for my new niece or nephew. I then learned that my brother and his wife may know the baby’s gender by December 17th. Did I want to wait that long to choose the colors and start?

Short answer: No. I got to thinking: It’s not like this won’t be a very multi-colored blanket. I had thought about using shades of blue or shades of pink at the beginning, but I don’t think that’s a good plan. Since the colors represent numbers, and since a baby’s going to see this, better to have distinct colors with distinct names as the prime factors that show up most often.

On top of that, I happen to have a full skein and more of a turquoise blue left over from another project. Turquoise worked out very well as the color for 2 in my prime factorization scarf. It doesn’t cry out “boy,” but neither is it a bad color for a girl. And best of all, it goes well with pretty much every other color. (And 2 has to do that.) I decided to go with bright, rich colors for the primes that will be most predominant.

Thanks to a fairly long management meeting and a day off today where I needed to read, I’ve already finished the first row. I’m very happy with the colors! Now, the first row consists of just knitting 9 squares. The next row of white will convert them into diamonds. I’m also proud to say that I sewed in the starting ends of all the yarn. And that’s my plan to go on with: At the end of each row, I’ll sew in all the ends that were loose at the start of that row. (I don’t want to sew in the ends right next to a live stitch.) That way, it won’t be a daunting task at the end of the blanket.

So here’s the first row, the numbers 1 through 9 (0 is a blank space.), with the numbers the colors represent written on the picture:

I admit it’s getting where I’m going to have a hard time giving this away! Good thing I’ve publicly said it’s for my new niece or nephew!

And that does remind me. If you’d like your own Prime Factorization T-shirt or Tote Bag, or if you have a friend or loved one who really needs one for Christmas, be sure to check my Cafe Press Prime Factorization Store! 🙂

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

Tweet

Alyssa Karise is here! And her blanket is ready to send! Here’s the finished blanket. It’s actually a rectangle, but I wanted the message to be in the picture, reading from the top to bottom, left to right, so the perspective warps it a little.

I used Base 5 math to code a message into the blanket, which I explained when I started. But now I can show you how it worked out.

You can see in the picture above that the blanket makes a sort of grid. The “smooth” squares in the blanket were knitted on the back side with the stitches P3, K1, P3 (purl 3, knit 1, purl 3). There are six of these squares in each row of the grid. I knitted my code on the back side so that the words would go from left to right. I coded letters into each square this way: P1, next two stitches are first digit of letter, K1, next two stitches are second digit of letter, P1.

I used two stitches for each digit of the letter, using a Base 5 code. I used P2 for 0 (since I was on the purl side); K2 for 1; yarn over, knit 2 together (ykt) for 2; ssk, yarn over (sky) for 3; and purl cable one stitch, holding to the back (cb) for 4. Here’s how the letters were made:

A: 01: p2 k2;
B: 02: p2 ykt;
C: 03: p2 sky;
D: 04: p2 cb;
E: 10: k2 p2;
F: 11: k2 k2;
G: 12: k2 ykt;
H: 13: k2 sky;
I: 14: k2 cb;
J: 20: ykt p2;
K: 21: ykt k2;
L: 22: ykt ykt;
M: 23: ykt sky;
N: 24: ykt cb;
O: 30: sky p2;
P: 31: sky k2;
Q: 32: sky ykt;
R: 33: sky sky;
S: 34: sky cb;
T: 40: cb p2;
U: 41: cb k2;
V: 42 – cb ykt;
W: 43: ykt cb;
X: 44: cb cb;
Y: 100: k2 p2 p2; (I knitted this as k2p1 (k1) p3, leaving the garter stitch in the middle.)
Z: 101: k2 p2 k2.

Here’s what I planned to have the blanket say (I added Alyssa’s name at the end when it was clear what that would be. Fortunately, I was knitting from bottom to top.):

Alyssa Karise,
Grace and Peace.
Grace and peace to you from God our Father and the Lord Jesus Christ.
Grace and Peace.

When I had the blanket all finished, ends sewn in, and I laid it out to take pictures — I discovered I’d left out a word! Urgh! But it’s still a Blessing Blanket. And I still thought about and prayed for Alyssa as I knitted. And it’s still warm and soft. I’m not going to say what word I left out, because I want to see if the baby’s parents can “read” it well enough to figure out! (I’m bad!) Astute readers of this blog who possess really really good eyesight might be able to tell as well.

To see how the coding actually looks, I took pictures of the top three rows. Here is AL – KA – GR:

Since the stitches are done on the purl side, k2 gives you straight bumps. So A = p2 k2 gives you a smooth panel, then bumps. L = ykt ykt gives you a hole from the yarnover, then two stitches together, in both sides. On the second row, K = ykt k2 combines both of those. Then we have A, which looks just like the first A. The third row has the same combination reversed in G = k2 ykt. Then R = sky sky. With sky, the stitches are knitted together before the yarn over, so the hole is on the right of the combined stitches.

The next section shows YS – RI – AC, and a fourth row, ND:

Y = k2 p2 p2, so I started on that first stitch I usually leave a purl stitch. So it looks the same as A, only shifted over one stitch to the left. S = sky cb shows us our final “digit”. The cable in back comes out as one stitch going over another with no hole. The second row is easier to see. R = sky sky, so you can see both sides have the hole on the right. Then I = k2 cb, and you can more easily see the cabled stitch crossing over. On the third row, we have A = p2 k2 and C = p2 sky. The fourth row, the end of the word AND, gives you a nice look at the cables again, with N = ykt cb and D = p2 cb.

I’ll give you the end of the top three rows, but I’ll leave it to the reader to work out that at least I didn’t make a mistake on the top of the blanket:

Making this was so much fun! In fact, I’ve been dragging my feet about sending it on. But tonight I noticed that the yarn label happens to have a pattern for a one-skein scarf, and I happen to have one skein left. So perhaps making myself a scarf out of this wonderful 85% cotton 15% angora yarn (Serenade) will remind me to send thoughts and prayers and blessings to my sweet little niece, Alyssa Karise.

And, meanwhile, my brother announced that his wife is expecting a baby. His baby needs a prime factorization blanket! I am swatching to figure out how best to accomplish this, and will definitely be letting my readers know!

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

Tweet

I’m a math nut. I’ve always said that my prime factorization sweater proves it.

I finished knitting the sweater in 2005. It shows the color-coded prime factorization of all the whole numbers from 2 to 100. I’ve shown it to lots of people since then, with mixed amounts of appreciation. In 2009, I finally posted a blog explanation of it and how it works.

Last April, I wore it to the US Science and Engineering Festival. I showed it to Ivars Peterson, manning the Mathematical Association of America booth. He tweeted one tweet with a link to my blog post, and the next day I got 28,000 hits on my website.

And the post got so much attention, this month it’s appearing in Hacker News! In print! I’m happy that people are finding out about it. It’s just goes to show that Math is Beautiful.

Now, there’s no way I’m ever going to make a replica of the original sweater. It took way too long. But since then, I’ve designed a t-shirt for sale to all on Cafepress. I’ve knitted a scarf that uses color-coded stripes to represent the prime factorization of numbers up to 50. And I’ve begun work on a cuff-to-cuff cardigan that also uses color-coded stripes, similar to the scarf. I hope it will go up to 100.

But for now I’m taking a break from knitting the prime factorization cardigan, because my little sister is having a baby!

I wanted to knit something special for the little one, but not so blatantly mathematical. I decided to use a mathematical code to knit a blessing into the blanket.

Here’s how it works. I’m using the same pattern I used to knit my son a cotton blanket 18 years ago. It’s a simple knit-purl pattern. I didn’t want lace, since I wanted it to function as a warm blanket. I gave it to him whenever he ate and managed to get him very attached to it!

I want to knit a code into the blanket. The layout is a basic grid. It mimics a plaid, but with knits and purls. I decided to focus on the stockinette sections that show up on one side. These sections are seven stitches across, with the stitch in the middle purled on every other row. There are six sections across in the grid and a total of 20 sections vertically down the blanket.

The baby is a girl, and her middle name will probably be Karise, coming from the Greek word for grace. I decided to use the Christian blessing that shows up many times in the New Testament: “Grace and peace to you from God our Father and the Lord Jesus Christ.” After all, what more could you want for a baby than grace and peace?

That’s 14 words, so I decided to put one word on each row of the grid, with the words “Grace and Peace” before and after, using all 20 rows. Each word is six letters or less, so I can use the six stockinette sections to highlight the letters.

Now, how to do the letters? Simple! Years ago, I did a library program where I showed the kids that they could use the ideas from the sweater to make codes using colors or shapes for letters. You start with a simple 1 to 26 code for the letters A to Z. Then I will convert the numbers to base 5 and I only need 5 different coded digits.

Here’s how the letters will be represented:

A – 01 B – 02 C – 03 D – 04 E – 10
F – 11 G – 12 H – 13 I – 14 J – 20
K – 21 L – 22 M – 23 N – 24 O – 30
P – 31 Q – 32 R – 33 S – 34 T – 40
U – 41 V – 42 W – 43 X – 44 Y – 100
Z – 101

Then I assign certain stitches to represent each digit. Since knitting is done from right to left, I’m going to work the stitches on the purl side, so when I look at them, they will read left to right.

Here’s the code I’m using, with two stitches for each digit.

0 = purl 2 (p2)
1 = knit 2 (k2)
2 = yarn over, knit 2 together (ykt)
3 = ssk, yarn over (sky)
4 = cable one purl stitch in back (cb)

Since the purl sections I’m using have 3 purl stitches, one knit stitch and then 3 purl stitches, I’m putting in the code this way: one purl stitch, then the first digit of the coded letter,
then one knit stitch, then the second digit of the coded letter, then one purl stitch.
I’m doing this on the middle row of the section of the grid.

I am working the code from the bottom of the blanket, so I had to list all the words in backwards order.

Here’s how it’s working out!

This picture shows the bottom three rows of the large grid. So I encoded the words “Grace and Peace.” Here’s the detail:

Looking at the first two sections of each word, on the top row, we have G = 12 = k2, ykt; R = 33 = sky, sky.
Second row has a blank section to center “AND,” then A = 01 = p2, k2.
Third row has P = 31 = sky, k2; E = 10 = k2, p2.

Here’s the middle section:

For the middle, on the top row, we have A = 01 = p2, k2; C = 03 = p2, sky.
Second row has N = 24 = ykt, cb; D = 04 = p2, cb
Third row has A = 01 = p2, k2; C = 03 = p2, sky

And finally, the last two sections in the row:

This just has the end of each word. On the first and third row, that’s E = 10 = k2, p2. We’ve got “blank” sections in the other slots.

Oh, and I should mention that on yarn.com, I purchased some fabulously soft yarn on sale. It’s Artful Yarns Serenade, and it’s 70% pima cotton, and 30% angora. There’s one strand of pink, one of purple, and one of brown, which I thought more interesting than straight pink. It’s ending up so beautiful and wonderful to touch. I always say that knitting with soft yarn is like therapy. And this time, I get to knit in a mathematical blessing.

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

Woo-hoo! After getting my prototype Prime Factorization t-shirt from my new Cafepress Shop, I tweaked the colors to make the chart brighter. I also made shirts on different colored backgrounds. As luck would have it, the new shade for 2 almost perfectly matches the “Caribbean Blue” background. My new revised shirt came this week, and I love it!

For those just coming in on this, the chart is 10 rows of 10, beginning at the bottom left. (Because I can’t bring myself to have higher numbers down lower on the shirt.) Each prime number gets its own color, and the prime factors of each number are displayed by color. Unlike the sweater, the numbers are listed on the shirt.

Like the sweater, I couldn’t resist putting rows of 8 on the back. And of course, I had to list those numbers in their Octal representation. But since that adds to the cost, I’m also offering t-shirts that have only the chart on the front.

Now, as long as I was so happily wearing my prime factorization t-shirt, I had to spend some time making progress on my prime factorization scarf. The biggest problem with the scarf is that I need to carry around lots of skeins of yarn. Fortunately, I ordered a Prime Factorization Tote Bag from my Cafepress Shop! (The pictured one was ordered before I brightened up the colors.)

I am super happy about how the prime factorization scarf is turning out. The original sweater is kind of pointless if you don’t know what it represents, but the scarf is aesthetically pleasing totally aside from the mathematics.

Here is my progress earlier this week, up to 24. (Today I finished 30, and, honestly, it’s hard to stop, because it’s so much fun to see how the next number will turn out!)

This one starts at the bottom. Though with a scarf, you can hold it any direction you like.

Once again, the background color, black, represents 1. It’s used between the other numbers.

The first stripe of blue is our first prime, 2.

Then we have pink for 3.

4 = 2 x 2. It’s not quite as easy to tell on the scarf, but the next blue stripe is twice as wide as the original.

5 gets a new color, yellow.

6 = 2 x 3, so a stripe with blue, then pink.

7 gets purple.

8 = 2 x 2 x 2, so that stripe of blue is three times as wide as the original.

And so it goes. I love the way the colors flow. You can quickly see that blue is showing up every second stripe, and that every second blue stripe is fatter. You can see the pinks showing up every third stripe and the yellows every fifth. As I said, I think it makes a lovely art object totally apart from the math behind it. And the math makes it completely cool.

Also, with a scarf, you don’t have to decide your colors ahead of time. I am using a certain progression. I’m afraid going to 100 is going to be ridiculously long, but I may do it anyway. After all, I’m doing it more for the math than anything else. But when I’m done, I’m planning to make a cuff-to-cuff cardigan using the same design, and that one cannot go so high.

I’ll continue to show my progress. Such fun!

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

Tweet