# Math Puzzles in Sewing: Lots and Lots of Trig (Or: Why Puzzles Matter and Formulas Don’t)

So, this conversation happened:

He asked for pics. They’ll involve some explaining, so a post is probably a better response than a tweet. Warning: this is long and has lots of pictures.

Math in sewing recently, in no particular order:

1.) Cross-stitching.

I do a ton of cross-stitching. The map (left) shows where to put each stitch on a grid. Different symbols represent different colors.

(The yellow squares on the map are the ones I’ve done already. You can see that I have a ways to go. Also, this is a gift for someone. Will that someone please forget that she ever saw these pictures? Thanks!)

I follow the map, and sew a small x onto the fabric where it’s supposed to go. The fabric itself is a grid with a bunch of holes at the corners. I try to do all of the stitches that I can with one color before moving on to the next color, because switching colors takes a while.

But, I don’t like wasting thread, so I want to optimize my path. Going to the closest spot that uses the same color may end up wasting a lot of thread. Which path uses the least amount of thread?

There’s also the question of making stitches themselves. You can see a stitch in the diagrams below. Red always comes before blue. But which order should I do the blue line in? Which one gets me closer to where I want to go next, and by how much? I use the Pythagorean Theorem about every two minutes when I’m cross stitching.

2.) Christmas Tree Skirt

I want to make a Christmas tree skirt. It’ll consist of sectors of a circle in alternating colors, red and white, and have a circle cut out of the middle where the tree goes. Something like this:

From my fabric, I’ll cut sectors of a circle, plus a little bit more for the seam allowances. But there are so many questions here: how many sectors should I have? I’ve drawn 8, but would 6 be better? Or 10? How much fabric of each color do I need to buy? Fabric typically comes in a fixed width, and you buy a certain number of yards, measuring the length.

But also – how much area will this cover? Will it have enough room to hold all of the Christmas presents? We don’t have children yet, so we only give each other and our extended family presents. That’s not a lot of presents. But when we have children, we’ll be giving them presents, and they’ll be giving each other presents – there’s a combinatorics problem in here. Then you have to think about the size of the presents. That’s a lot of unknowns, so there was no way I could calculate the area I’d need to hold the presents. But I could and did calculate the area of the circle, to make sure that we’d have a reasonable amount of room.

This was my scratchwork for this problem:

Stacking sectors together the way I’ve drawn them minimizes the wasted fabric. The width of my fabric is 44, so that’s the radius of my circle. To figure out the length of the fabric, I need the length of the chord that cuts across the circle; that’s a simple use of sine. Then I need the distances between the central radii of the adjacent sectors.

I’ll be honest: I don’t remember exactly how I solved this, because I was working on it in November. Clearly, I should have shown more work, instead of just writing down my answer. If I was doing it this morning, I’d probably draw a ton of right triangles and use the Pythagorean Theorem. I distinctly remember using either the Law of Sines or the Law of Cosines, and I remember that making it easier, but I’m just not seeing it right now. Sorry, Dan. But this was definitely the problem where I used it.

3.) Eight-pointed stars

My fabulous and talented mother made me this:

I think it’s amazing and want to make a lot more. I want each banner to have different stars. There are a ton of ways you can combine squares and triangles to make eight-pointed stars:

If I need the entire design to be a certain size, how large should I cut the fabric? It’s hard to cut out a star, so what a quilter typically does is cut out the shapes that make it up (plus a seam allowance) and then sew them together. That also lets you play with different colors.

Some figuring I was working on for this, with some (rounded) results of the Pythagorean Theorem:

4.) Mariner’s Compass

This is a Mariner’s Compass:

This is a particularly lovely one, and you can read all about how Mary MacIlvain designed it here. I’m hoping to make one this summer. But this leads to so many questions: how big should it be? What are all those angles and lengths? There’ll be a lot of trig here.

5.) Six-pointed Star

My mother wanted to sew a Star of David with embroidery floss on cross-stitch fabric. That fabric has a grid on it, so she sketched out a map on graph paper.

She was having trouble drawing it onto a grid and getting it to where the points lined up with holes in the grid. We sat down together, said that one of the sides had length $a$, and proved (with the Pythagorean Theorem) that it was impossible. You always get an irrational number floating around, so if she wanted a perfect six-pointed star, she was going to have to go in between holes. That’s not that hard from a sewing perspective, but our two-minute mathematical adventure saved her about an hour of re-drawing the silly thing.

The other nice thing about that is that my mom used to be a middle school math teacher and she now tutors Algebra I, so she was hanging right in there with me when we were proving things. That’s right – my mom makes me quilts and does proofs with me. What more can a girl ask for?

So, math in sewing. A lot of it boils down to the Pythagoream Theorem and trig ratios. Sometimes combinatorics gets involved. None of this was contrived either – these really were questions that I’ve had to answer over the last few months, or questions that I’ll need to answer in the next few months. (I know that’s not this week, but I haven’t had time to do much besides cross-stitching lately. That’s the end of the school year for you.)

Now, after catching a lot of flak from teachers who weren’t sure where he was taking this conversation, Dan tweeted this:

(For those of you who don’t tweet, read bottom up.)

I completely agree with him. I mean, I teach BC Calculus, among other things. Most of that is rarely used in daily life, even for engineers. (My husband, who is an engineer, hasn’t found the area of a polar function since college.) But Calculus isn’t beautiful because it’s useful; it’s beautiful because it’s interesting and it’s fun and it’s a puzzle. I teach my AB kids to find the area under curves by using rectangles. Now, here: how will you find the area under this curve defined by a polar graph? Rectangles aren’t going to be easy to use; what will you do instead?

Sure, I used a bunch of high school math in my sewing projects. I’d argue that kids need to have at least geometry for the hard-skills, because we need to deal with trig ratios, the Pythagorean Theorem, and triangles and circles and such. Those are useful. But if it was just about the hard skills and formulas, we could probably force those down into middle school, not worry about the understanding, and call it a day. And I can do all of this without math; it just involves a lot more trial and error and trips to the fabric store.

See, the point isn’t that I used a bunch of high school math. The point is that I had a puzzle that I needed to solve, and I knew that math was a tool that could help me solve it well. I have a master’s in math, not math education. I don’t use the concepts I learned much. There’s just not a lot of call for discussing topologies and complex analysis with high school students (though goodness knows I’ve tried!)

But my math education is by no means wasted. I learned a lot more than how to prove that something’s a topology. I learned about breaking problems into smaller bites, about thinking logically, about proving things, and about playing with ideas just for the joy of it.

And really, that’s why I teach math. I’d be shocked if any of my kids needed to prove a trig identity as an adult. But all of them will need to be able to make a logical argument at some point in there lives. They’ll need to be creative in a non-artsy way. They’ll need to solve a puzzle. I don’t know any subject that lets students play with those things as much as math does.

Because that’s what math is: not formulas and procedures, but puzzles and ideas. And if I sell math to my students by saying, “You’ll need this later,” I’ve robbed them of the beauty and the joy that I’m supposed to share with them. I need to grow in that, but it’s something I’m working on.

So, Dan, yes, I did use a bunch of trig in my sewing, and I promise that the Law of Sines or Cosines was in there. But your point still stands – it’s not the hard skills that were useful here, but the joy of finding and solving a puzzle.