Prime Progress

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.

Prime Factorization Swatches

I finished three swatches and decided on how I’m going to approach my new Prime Factorization Scarf.

First, I tried garter stitch. It’s a reversible stitch, and I thought it would work for the scarf. No pun intended, I thought I’d begin with the primary colors, since they are clearly “red,” “blue,” and “yellow.” Here’s how that swatch turned out:

The colors do stand out in this version. Black is 1, the background color. The pattern starts on the left. 2 is red. One ridge clearly stands out for the factor. When you get to 1 again, you stop multiplying and start a new number. Primes will always be one ridge. Next is 3, which is blue. Then comes 4, clearly showing as two ridges of red, 2 x 2. Then black, so you start a new number. Then comes 5, a new color, yellow. Then black. Then we have a ridge of red with a ridge of blue, 2 x 3 = 6. A ridge of black. Then 7, a new color. Black. Three ridges of red, 2 x 2 x 2 = 8. Black. Two ridges of blue, 3 x 3 = 9. Black. A ridge of red and a ridge of yellow, 2 x 5 = 10.

What I like about this swatch? The colors pop out and clearly show how many of each factor you want.

Problems with this swatch? Well, I completely forgot that garter stitch with colors is not reversible. Here’s the other side:

The garter stitch pattern would work great in a sweater, where you only see one side. But a scarf is better when it’s reversible, since it’s not easy to keep only one side of a scarf showing.

One other problem was that I didn’t like the colors. In the picture on the web, the red looked like a pretty cherry red. In real life, it was so bright, it almost glowed in the dark. Next to black, it feels glaring.

So I decided to experiment with the colors I think are the prettiest of all the ones in the box. After all, the color for 2 is going to get used in every other stripe. Might as well use a color I like to look at, right?

I needed to pick a stitch that would be reversible in color. I chose the Double Seed Stitch, which gives a nice textured look. You alternate two knit stitches with two purl stitches, and you do two rows that line up with each other. Then you do two rows with the stitches staggered from the first two — The knit stitches in the two top rows are lined up with the purl stitches in the bottom two rows, and the purl stitches in the two top rows are lined up with the knit stitches in the bottom two rows.

You can’t really tell the colors I used in the next swatch, but it’s a turquoise for 2 and pink for 3, yellow for 5, and purple for 7. I got happier and happier as I knitted. I think this is beautiful, and it will be the pattern I use for the scarf. Here’s that swatch:

Now, I acknowlege that it’s not nearly as easy to tell exactly where the numbers separate or exactly which factors are in the number. But it’s pretty. And I know which factors are where. And it’s completely reversible. The back looks exactly the same as the front.

I did one more swatch, to be absolutely sure I wanted to use a black background. I also wanted to see if using larger needles would make the fabric more drapey. The yarn band suggests size 8 needles, and that’s what I’d been using, and I think it would be nice for a sweater. But for a scarf, I’d like a little less firmness. So I made the next swatch with a cream-colored background and size 10 needles, but the same stitch and the same colors for the numbers. Here’s that swatch:

Yes, this combination is pretty. But I think it reminds me too much of my original prime factorization sweater, and I’m ready for something different. So I’m going to go with the black background.

So I’ve begun! I’m using Size 10 needles, a black background, double seed stitch, and 26 stitches across. (The swatches used 20 stitches.)

Oh, one more thing: I hereby resolve that I will, I WILL, sew the ends in as I go, so that I don’t have to do them all at the end, like I did with the sweater. (Sigh. The worst thing about this whole project!)

I also plan to make a black edging along the sides, to hide the carried-along yarn. I can practice on the swatches.

And while I’m knitting, my mind will be spinning with ideas for a cardigan.

Stay tuned and watch my progress! One of the fun things about it is that I can wait to choose colors as I go. I don’t even have to stop at 100….

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

Ready to Start My Prime Factorization Scarf!

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.

Prime Factorization Knitting Revisited

Yesterday I went to the US Science and Engineering Festival in DC, and made sure to stop at the Mathematical Association of America booth. I knew they’d be there because every day I make sure to do MAA Minute Math.

I was hoping it would be cool enough (weatherwise) to wear my prime factorization sweater (For the explanation, follow the link!), and to my joy it was. I was happy to get a picture taken at the MAA booth.

Well, that got the attention of many more math people, and today I found four new comments on my blog post about the sweater and a page about me on Hacker News!

Now, the good people at Hacker News did misread my age, so I will post something I just realized that will be true this year after my younger son and I have our birthdays. (I was so delighted when I realized it, I ran to tell my son, not realizing he’d just gone to bed.)

My age will have five prime factors.
My oldest son’s age will have four prime factors.
My youngest son’s age will have three prime factors.

A picture of the three of us should enable you to narrow it down. (I am not 72, and my youngest son is not 8.)

I’ll add one more cool set of facts to definitely set our ages:

In the year my oldest son was born, my age had four prime factors (as his does now).
In the year my youngest son was born, my age had three prime factors (as his will this year).

Let’s see. My youngest son’s age is still ambiguous. So I’ll add the clue that there are only three distinct prime factors in all the expressions above. That should do it.

So, all this establishes that I was thinking about prime numbers yesterday. And, yes, I think it’s time to make a prime factorization cardigan, which I can wear in warmer weather.

Some have said I should sell these. But let’s be honest. Having to buy all those shades of yarn costs around $100. Then it was definitely not my only knitting project, but it took me more than a year to knit. (Fun time, but not worth spending if it were for monetary gain and not for the fun of it.) Then it took me more than a month just to sew in all the yarn ends. Granted, if I made more with the same color scheme, that would spread out the cost. But by the time I finished, I’d had quite enough of the project. It has taken ten years for me to be at all interested in doing anything similar, and I’m NOT going to use the same yarn and color scheme as before; you can be sure of that.

With a cardigan, you wouldn’t have room for a chart on both sides. So I was thinking about how else to express it. Last night, I ordered some Plymouth Encore yarn from yarn.com to make a scarf.

Why Plymouth Encore? Well, first, it’s on sale right now. Needing 26 different colors to go up to 100, any savings per ball helps. I also decided that my cotton sweater (the original was made in Cotton Classic), while soft and comfortable, is a little bit droopy, and wasn’t the greatest choice for the intarsia work. Wool by itself risks being too scratchy. Most of all, this had enough colors, which I hope will be distinct.

This is my plan this time:
I think I’ll use a black background this time. I don’t look good in black is the one reason I didn’t use that on my sweater. (The original partial picture of a blanket that gave me the idea used a black background.) But in a scarf, the background won’t be as prominent.

I’m thinking I’ll use garter stitch, with two rows and one ridge for each factor. I will probably put two stitches of black (one) along the sides, in order to (I hope) hide the two and three factor colors being carried along the edge of the sweater. (I absolutely WILL sew in ends as I go this time. I will. I WILL!)

So here’s how it will work. I’ll start with however many rows of black looks good at the beginning. This is 1. Then I’ll choose a color for 2 and knit two rows (one ridge) in that color. Then I’ll do two rows of black. That represents a factor of 1, and also tells you that we’re starting on a new number. 3 will get two rows of a new color. Then two rows of 1. 4 is 2 x 2. So 4 will be expressed with four rows (two ridges) of the 2 color. Then two rows of 1. Then a new color for 5. Then two rows of 1. Then 6 = 2 x 3, so two rows of 2 directly followed by two rows of 3. Then two rows of 1.

Get the idea? Numbers with lots of prime factors will take up more space than prime numbers.

Mind you, I’ll swatch it out, with some different color choices, and see what looks best. (You definitely want your favorite colors as 2 and 3.) I will post pictures when I get there.

Once I have a scarf done? Well, what I might try with a cardigan is a chart like the old sweater on the back (in new colors and yarn) and maybe stripes as in the scarf on the front sides. Or maybe I’ll be sick of it and give it a rest for awhile.

Based on the new comments, it’s time for me to design a t-shirt! Stay tuned. I have written a children’s book which I called Colors and Codes that talks about using these ideas to make cyphers and patterns using colors or shapes combined with math. As part of the book, I made several charts on my computer. I will see how hard it is to transfer these charts to a t-shirt in Cafe Press.

By the way, I haven’t tried to sell this book yet. I had been working on selling a middle grade novel, and lately I’ve been letting both efforts rest while I dealt with some medical issues. But if anyone knows of an agent or a publisher who’d like to take on something a little unorthodox but extremely cool to math geeks, let me know!

It’s been about ten years since I designed and knitted the sweater. (So I was WAY young then!) Let me stress that the idea of visualizing the prime numbers through colors in knitting was not my own. By all means, spread the word! The article I read (and I should definitely track it down. It was in Interweave Knits in the late 90’s or 2000 or so.) talked about how the blanket that had been made inspired kids who didn’t think they were good in math. As I say in my book, you can attach the numbers to the letters of the alphabet and use these ideas to knit or color messages into things. The sky’s the limit, and it’s lots of fun.

Once I have some swatches, I’ll take some pictures and post the results!

Edited to add: I found the inspiration! It was an article in the Fall 2003 issue of Interweave Knits, called “geekchic” by Brenda Dayne, regarding the work of Pat Ashcroft and Steve Plummer. They have a fabulously cool website at woollythoughts.com. Here’s what the article said about an afghan they created:

“Across the Atlantic Ocean and far from the research laboratories and hallowed halls of Academia, a young girl, age thirteen, stands mesmerized in front of a knitted afghan displayed at the annual North-East Math Fair in Lancashire, England. Constructed of one hundred brightly colored squares, the intricately striped fabric is the creation of Pat Ashforth and Steve Plummer (www.woollythoughts.com). Knitters, teachers, mathematicians, and partners, Pat and Steve have found that basic mathematical principles make for beautiful knitwear designs, and that knitting is an excellent way of explaining complex theorems to their students.

“Vibrating with color, and reminiscent of African Kente cloth, the Counting Panes afghan is so beautiful it’s hard to accept that it was created as a teaching tool. Within its one hundred brightly colored squares, in ten columns and ten rows, however, lie lessons in multiplication, division, pattern, and numerical relationships… If a square contains yellow, it contains a number divisible by two, if it contains red, then the number divides by three. The more colors in a square, the more numbers it divides by.”

Now, they only had a photograph of a very small part of the afghan, so I couldn’t see how it worked. That part would not work on my sweater at all — there are several squares that appear to just be dark blue, and it’s stated elsewhere that green always appears with baby blue. So maybe that quilt is just showing factors of numbers, and not the prime factorization? Maybe beyond a certain point primes don’t get new colors?

But anyway, having the fondness for math that I do, that much information made me realize that I could knit a prime factorization chart. But I wanted to wear it, not just look at it! I still can’t make sense of what, exactly, their afghan was doing, but they are the ones who gave me the idea behind my design. I graphed it out, figured out how many stitches across I needed, and then found a basic sweater pattern from the book Picture Knitting to use as my canvas. Thank you so much for the germ of the idea!

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

Review of Casting Spells, by Barbara Bretton

Casting Spells

by Barbara Bretton

Berkley Books, New York, 2008. 308 pages.
Starred Review
2010 Sonderbooks Stand-out: #6 Fiction

Happy Valentine’s Day! Today I’m appropriately posting the review of a romance, but this one is a paranormal knitting mystery romance! The combination is delightfully original and a whole lot of fun.

Here’s how we meet Chloe Hobbs:

“By the way, I’m Chloe Hobbs, owner of Sticks & Strings, voted the number one knit shop in New England two years running. I don’t know exactly who did the voting, but I owe each of those wonderful knitters some quiviut and a margarita. Blog posts about the magical store in northern Vermont where your yarn never tangles, your sleeves always come out the same length, and you always, always get gauge were popping up on a daily basis, raising both my profile and my bottom line.”

Chloe’s store seems magical because it is. She’s the daughter of a sorceress who fell in love with a human. But her father died in a car crash when she was six years old, and her mother chose to leave this world to be with the man she loved. Chloe inherited several things from her mother including a basket of roving that remained full to overflowing no matter how many hours she spends spinning it into yarn. But she also inherited a responsibility to the town.

“Over three hundred years ago one of my sorcerer ancestors cast a protective charm over the town designed to shield Sugar Maple from harm for as long as one of her line walked the earth and — well, you guessed it. I’m the last descendant of Aerynn, and if you thought your family was on your case to marry and produce offspring, try having an entire town mixing potions, casting runes, and weaving spells designed to hook you up with Mr. Right.”

Unfortunately, the protective spell seems to be weakening. And there’s more than just protection from accidents and crime at stake. Because Sugar Maple “wasn’t the picture-postcard New England town our Chamber of Commerce would have you believe, but a village of vampires, werewolves, elves, faeries, and everything else your parents told you didn’t really exist.” However, Chloe’s mother really came into her powers when she fell in love, so maybe that’s all that Chloe needs.

But then a visiting beautiful stranger dies. The first tourist or nonvillager ever to die within town limits. Aerynn’s spell is definitely waning, because that’s not supposed to happen.

Sugar Maple doesn’t have any police force, since it doesn’t have any crime. So a policeman from Boston, who knew the deceased, goes up to the scene of the crime to investigate.

What follows is funny and quirky and full of surprises. Can the whole town hide the truth from him? And what will happen to the town if the spell fails? What will happen to Chloe?

I must admit, the romance is not exactly subtle. As Chloe begins to have magick, it basically throws her into the guy’s arms. But it is humorous to read about her trying to explain it!

This book is a light-hearted romp through a most imaginative situation. Definitely the best paranormal-romance-knitting-mystery I’ve ever read! And there are knitting tips at the back! How can you go wrong?

Buy from Amazon.com

Find this review on Sonderbooks at: www.sonderbooks.com/Fiction/casting_spells.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Source: This review is based on a library book from the Fairfax County Public Library.

My Prime Factorization Sweater

pfsweater

I wore my Prime Factorization Sweater to KidlitCon09, and it shows up in all my pictures, so I think it’s time for me to explain it.

This is the sweater that proves that I am a Certified Math Nut.

Okay, here’s how it works. You have to start in the bottom lefthand corner, because the mathematician in me couldn’t bear to start anywhere except where the origin would be on Cartesian coordinates. Naturally, the numbers go from left to right and from low to high.

I’ll post a picture of the front of the sweater:

pfsweater_front

Okay, look at the bottom row. It looks like there is a blank space on the left. That represents 1, because 1 is the background color, because 1 is a factor of every number.

Next is a blue square, which represents 2.

Next is a red square, for 3.

Then comes 4. 4 = 2 x 2. So 4 is represented by two blue rectangles.

Then comes 5. 5 is prime, so 5 gets a new color, yellow.

Next is 6. 6 = 2 x 3. So 6 is represented by a blue rectangle and a red rectangle.

7 gets a new color, purple.

8 comes next. 8 = 2 x 2 x 2. So 8 is in a square with three blue rectangles.

Then comes 9. 9 = 3 x 3. Two red rectangles.

Last on the bottom row is 10. 10 = 2 x 5, so we have blue and yellow.

The second row starts with 11, which is given the color pink.

12 has three factors, since 12 = 2 x 2 x 3, so two blues and a red.

Get the idea? This sweater presents a chart giving the color-coded prime factorization of every number from 2 to 100.

The patterns are wonderful and fascinating. You’ll quickly notice that the yellows and the blues line up, because 5 and 2 are factors of 10. You also might notice that all perfect squares are symmetrical. Multiples of 11 go in a lovely pink diagonal across the sweater. There are hundreds more patterns. It would be a lovely visual aid for teaching number theory. Fun to quietly wear to Math competitions, too!

What’s more, you can use this as a quick conversion table to convert to Octal (Base 8), because on the back I did the same thing with rows of 8:

pfsweater_back

The fun thing about rows of 8 is that the patterns are all different! Notice how the last column is full of blue squares because every number there is a multiple of 8 and has at least three factors of 2. And now 9 (two reds) acts like 11, going diagonally up the sweater, as does 7 (purple) in the opposite direction.

On the sleeves, I did rows of 2 and rows of 3. The rows of 3 is the only one where the blues do not line up, because 2 and 3 are relatively prime.

Isn’t it just the coolest thing in the world?!!!

Okay, I warned you: This is the item that proves I am a Certified Math Nut. I can get hugely excited and animated talking about this sweater.

I have already done a library program called “Puzzles and Patterns” showing kids how they can make simple codes using the ideas from this sweater. There’s definitely a children’s book in there, but I haven’t gotten around to writing it yet. I definitely plan to some day!

One of the cool things about this sweater is that it works in any language and on any planet!!! You see, even if an alien race had only four fingers on each hand, they could look at the back of the sweater and all their numbers would work. For that matter, a number system with a base of 7 or some other strange base would still work, even though it might not be in neat rows for that base. The chart is entirely independent of the symbols used to represent a number, and based only on color.

So we had a family joke that if an alien ever came to our door, we’d run and get the sweater to prove that we are intelligent life.

I only hope the aliens are not color blind!

Of course I also like to tell the story that when I was knitting this sweater, I brought it along to visit my family and friends one Christmas. Most of my family are Math Geeks, too, so they were impressed. But one friend had a young son who listened to my explanation and responded, “That’s just weird!

What can I say? He does have a point. Call me weird, but I still think it’s one of the coolest things in the world!

Edited to add: Here’s a link to my CafePress store, where you can order t-shirts using this idea, showing the color-coded prime factorization of the numbers from 2 to 100, with the number also printed below the color-coded square.

Also, here are all my entries about other prime factorization projects.

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