Archive for the ‘Prime Factorization’ Category

My Pascal’s Triangle Shawl

Thursday, February 13th, 2014

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!

Pascal’s Triangle Shawl to Row 10

Thursday, November 28th, 2013

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 yarn.com, 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!

More about Mathematical Knitting

Saturday, August 24th, 2013

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
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!

The Prime Factorization Blanket!

Wednesday, August 21st, 2013

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!

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.

Halfway on the Prime Factorization Blanket!

Friday, April 12th, 2013

I’m halfway finished with my new niece’s Prime Factorization Blanket!

That’s the good news. The bad news is that it doesn’t look like I’m going to finish before Arianna arrives. But the good thing about a blanket is that it doesn’t have a size, right?

I actually finished up to 49 a week ago. I did not take a picture and report my progress. And then — when I started on the next row and began knitting 50 — I discovered I had used the wrong color when I knitted 40! I had used the color for 3 in place of the color for 5! *shudder*

(Those who need to be brought up to speed, I explain the blanket in previous Prime Factorization posts.) 40 should be 2 x 2 x 2 x 5, so the square was divided into four sections, with three of them the color for 2 (blue), and one the color for 5 (green). But, horror of horrors, I had used the color for 3 (yellow)! And I didn’t even discover it until I after I had knitted 50 = 2 x 5 x 5, also using yellow when I should have been using green! Yikes! But then I was getting ready to start on 51 = 3 x 17, and then it dawned on me that the color for 3 is yellow, so it is NOT the color for 5, and I’d been doing it wrong!

Fortunately, it was on the end of the row, and relatively easy to fix. I took out the knitting I’d done on 50, undid the last white square in the row, and then undid just the yellow section I’d put on the top of 40. I replaced it with green, knitted the white square back, and then started up on 50.

But I’ve decided that taking a picture after each row and *checking* the numbers carefully is a good idea!

I did discover a mistake when I finished knitting my prime factorization sweater. I think it was in the rectangle for 48. But 48 has five factors, and in that piece of knitting the factor for three only was about four stitches. So I was able to pick them out and put in the correct color with a crochet hook! It would not be so easy to do on this blanket, so I am going to have to be more vigilant!

Librarians Help – With Math!

Tuesday, March 26th, 2013

Today I had my Colors and Codes program that I mentioned last week.

Now, I spent ten years of my life teaching college math, but doing math programs at the library is so much more fun!

Why? Well, the biggie is I don’t have to grade them, so it makes the whole thing much more light-hearted. I’m showing them things about math that I think are really cool, and they get to think about ways to do it themselves. And it’s all just for fun. At the library, we teach people things they want to know! If they don’t want to know them, they don’t need to come. It’s that simple!

Here’s what I did. I showed the kids my prime factorization sweater (wore it of course), and we worked out how it works. (That was fun!) I told them if colors can represent numbers, they can also represent letters. Just use 1 to 26 for A to Z. So you can write messages this way. I showed them a prime factorization code, then showed them other bases and how you can make codes with them. We wrapped it up by getting out sticky foam shapes and they could put a coded message or just a pretty pattern around a picture frame or on a bookmark or a door hanger.

The highlight for me, I think, was when a girl was working on coloring in the prime factorization chart on the hand-out. She was stuck on 24. I asked her what it equaled, and she said 12 x 2. So we looked at 12, and then the light went on and we talked about how you could do figure it out different ways, but you always got three 2s and one 3.

Now, I’m going to write some notes to myself while the program’s fresh in my mind. It went well; the kids had fun. But I want to do it again this summer, and hope it will go even better.

1. I’ll set the age level higher. I do think I lost a few kids this time. I think I’ll set it at 10 or 11 years old rather than 8. You want the kids to be fully comfortable with multiplying. Now that I think about it, when I did this program a few years ago at Herndon Fortnightly Library, I think the age limit was 10.

2. We’ll do some coloring on the prime factorization chart before I move on. This group did work out with me how it works. I didn’t want to get bogged down, but I think some coloring would help them understand it better.

3. I’ll have them figure out the numbers for their name in every code I go over. For example, my name, Sondy, in a base 10 code is 1915140425. (S is the 19th letter, O is the 15th, and so on.) In a prime factorization code, it’s 19 1 3 5 1 2 7 1 2 2 1 5 5 1. (19 x 1, 3 x 5 x 1, 2 x 7 x 1, 2 x 2 x 1, 5 x 5 x 1) In a Base 6 code, it’s 3123220441. In a Base 5 code, it’s 34 30 24 4 100. In Binary, it’s 10011 1111 1110 100 11001. Taking the time to do that would mean they’d get what I was having them do when they went to use the foam sticky pieces.

4. We’d do some coloring on the other charts before we moved to the foam shapes. Then I’d have them do their name with the colors they picked.

5. I’d show them exactly how I did my name on the bookmarks, one using colors and one using shapes.

Did I mention everyone did have a good time? But I think I’ll do a little more hands-on, using their names, before I move to the craft next time.

But it was a great trial run!

And don’t forget! Librarians help! We get to show kids how much FUN Math is! And we don’t even have to test them on it!

Colors and Codes

Tuesday, March 19th, 2013

I just got a tweet that made me prouder than I’ve EVER been of my Prime Factorization Sweater, and that’s saying a lot.

The tweet was from @milesmac, Miles MacFarlane, a teacher, with the words, “#LeilaN students deciphered @Sonderbooks Prime Factorization Sweater – Now making own code #7Oaks”

Here’s the picture that accompanied it. Even by the backs of their heads, you can tell those are engaged kids!

Yes! That’s what it’s about! Mr. MacFarlane, you made my day!

And the timing is lovely. Next week, at my own City of Fairfax Regional Library, I’m doing a program I’m calling “Colors and Codes” where we’re going to do exactly that. I’ll wear the sweater (or maybe my prime factorization t-shirt and bring the sweater. And the scarf). I’ll show them how we can assign each letter of the alphabet a number from 1 to 26. We’ll start with a factorization code, but move on to things like Base 6 or Binary. And I’m going to have foam shapes for them to make crafts with codes in colors or shapes.

Yay! See, we don’t have to make Math fun! Math is fun!

Prime Factorization Progress – To 39

Tuesday, February 26th, 2013

I’ve already posted several times about my Prime Factorization Knitting, and I can’t resist posting pictures every time I get another row of numbers done on my new niece’s Prime Factorization Blanket.

You can get more detail of how it works in the earlier posts, but basically each prime number gets a color, and each number gets a square divided into the colors for the factors of that number. I’ve finished up to 39. (I’m not putting an exclamation mark after that statement, since I haven’t gotten to 39 factorial.) Here’s how the blanket looks so far:

And here’s a close-up on each side, with the numbers written in. You’ll have to figure out the factors. And I can assure you that it’s a lot easier to tell when there are two or three (or four or five) of the same factors in one number when you can see and feel the blanket. I divided it with garter ridges, and the photo couldn’t really catch that.

Here’s the left half:

And the right half:

Don’t forget that you can get your very own Prime Factorization T-shirt at my Cafe Press shop for a lot less effort than this blanket is taking (but okay, you won’t have as much fun as I’m having). I took it to a Youth Services Librarian meeting today, and only the unwary asked what it was going to be. I must admit, it’s a lot better for knitting during meetings when I’m on one of the white rows.

Will I finish before Baby’s arrival in May? I hope I will at least be close….

Prime Factorization Blanket – to 29

Monday, January 21st, 2013

I got done another row of numbers on the Prime Factorization Blanket for my arriving niece!

It’s hard to see the ridges in the solid colors, so here are close-ups of the left half, then right half:

The bottom row in the picture is 1 (white), 2 (blue), 3 (yellow), 4 = 2 x 2, 5 (green).

The second row is 10 = 2 x 5, 11 (red), 12 = 2 x 2 x 3, 13 (tan), 14 = 2 x 7, 15 = 3 x 5.

The top row is 20 = 2 x 2 x 5, 21 = 3 x 7, 22 = 2 x 11, 23 (baby blue), 24 = 2 x 2 x 2 x 3, 25 = 5 x 5.

Now the right half:

Here we have the bottom row of 5 (green), 6 = 2 x 3, 7 (dark purple), 8 = 2 x 2 x 2, 9 = 3 x 3

The second row is 15 = 3 x 5, 16 = 2 x 2 x 2 x 2, 17 (pink), 18 = 2 x 3 x 3, 19 (dark pink).

The third row is 25 = 5 x 5, 26 = 2 x 13, 27 = 3 x 3 x 3, 28 = 2 x 2 x 7, 29 (periwinkle)

I really like the way it’s turning out!

You can read more about my prime factorization knitting in previous blog posts or via my Pinterest board. And don’t forget to look in my cafe press shop for prime factorization t-shirts.

Prime Factorization Blanket – Second Row

Friday, December 21st, 2012

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….