64th anniversary tablecloth II: practice

And now, the exciting conclusion of the tablecloth saga!

I should just say that this project was harder than I thought it would be. In theory, there's no difference between theory and practice. But in practice, there is. One lesson I learned was that the lumps and bumps of the human body are a lot more forgiving than a flat tabletop. Who knew?

My grandparents' anniversary was September 5, the day before my birthday. (I was a 35th anniversary present, a day late.) It's not really fall weather in Dallas, but my grandmother likes fall colors, and their house color scheme is fall-ish, so I went with some orange eyelet I had lying around and brown and cream polyester for the tablecloth. After some sketching, I decided that the color scheme, working in from the largest square, should be orange-white-brown-white-orange-white-brown-white.

When I left off the last post, I had just started cutting the fabric. I was a little worried about having enough of brown and orange, so I figured out a super clever strategy: the smaller squares covered up a lot of the larger ones, so I poached fabric from the larger squares to make the smaller ones of the same color. For the white ones, I cut the two larger squares out separately and then cut the 3rd largest out of the fourth largest.

I also figured out a super-clever way to cut a small square out of the center of a large square: fold the large square in half to get a double-layer isosceles right triangle. Then fold again to get a four-layer isosceles right triangle. Then, keeping measuring tape parallel with side of outer square, move it in until you hit the side length of the small square. (In the schematic on the left, I start with the full square and fold it up into the small triangle. The dotted line in the last triangle is the side of the smallest square.)

Unfortunately, I once again underestimated the difference between theory and practice. It's very hard to get all the layers perfectly flat, and some of the squares I ended up cutting out weren't very square. The large orange one worked out OK, but I had to re-cut some others out of the remaining fabric, which didn't end up being an issue. Oh well, it was a nice idea.

I also messed up the largest white square. I don't know how, but it ended up with sides that were two inches too small. I didn't have a square of fabric large enough any more, so I ended up cutting out four right triangles, the only part visible in each layer, instead of one square.

This is the orange piece with a square cut out of the middle. That white layer is the one I had to make by cutting out triangles instead of a square because I miscalculated the first time.

The construction was pretty straightforward. I started from the innermost square and worked my way out. Instead of doing some sort of turn-under on the sides of each square, I topstitched using a wide but short (many stitches per inch) zigzag stitch. A wide, short zigzag stitch can cover a multitude of sins! I was nervous about the (lack of) square-ness of the squares, and the wide stitch gave me enough wiggle room to make less than straight sides look straight. (If you'll recall from the last post, I had computed the side lengths to the nearest thousandth, and here I was using a quarter-inch wide zigzag stitch-ha!) For each layer, I chose the color that was not represented by the two adjoining squares.

I had some trouble getting the fabric to lie flat. I had to pin a lot more densely than usual, and I also basted the layers. (I'm lazy, and I don't usually baste when I'm sewing.) In the end, I didn't completely get it to lie flat, but it was pretty good.

I went to Dallas for a week in September to be with my mom when she had surgery, so I decided to deliver the tablecloth in person then. Of course, I procrastinated and was trying to finish it the night before I left. Of course, I ran out of orange thread because the zigzags were so thread-intensive, so I had to run to the fabric store the next morning to pick up more. I had time to finish all the zigzags, but I didn't have a chance to do some of the finishing.

The upside to my procrastination was that I got to use my mom's beautiful 1950s-era Singer sewing machine to finish it off. (We think it belonged to a grandmother/great-grandmother/great-aunt, but Mom can't remember exactly how it got passed down to her.) Before I had a machine of my own, I used this one, and now that I am more experienced, I have a better appreciation for what a great machine it is. I had to get her to help me remember how to thread it and insert the bobbin, but once I had it all set up, it was a dream. I don't know how to describe it exactly. It's just a smooth ride. Is there a certain car that's supposed to have a really smooth ride? A Porsche or something? I've never driven a Porsche (or whatever the canonical smooth ride car is), but if I had, I would compare using this machine to driving a Porsche. I love my machine, and it can do zigzags and buttonholes better than Mom's, which requires special attachments, but I wouldn't mind having a machine like hers for straight stitching someday.

Since I had a big hole in the back of the piece from my weird inner square cutting, I used a big piece of leftover white fabric to make a backing for the piece. Here it is before I covered the hole. (I didn't take a picture after. Oops.)

My mom sent me some pictures of the finished tablecloth on my grandparents' table, one of which is posted at the top of this post. I hope they like it. I love and admire them so much, and I hope that our marriage can be as long and as loving as theirs. I often have trouble finding good gifts for them, so I was glad to get to make them something special and creative.

Jon LOVES the design. He's supportive about my sewing, but he's not usually very effusive, and he doesn't often bring a design up later. I might be able to make a present for him along these lines, although I'll probably go smaller/fewer layers. Wrangling eight was a bit overwhelming.

64th anniversary tablecloth I: theory

In my last post (two months ago!), I asked for ideas for a tablecloth that would feature 8 squares for my grandparents' 64th (8 sqared-th) anniversary. A twitter friend pointed me to Wooly Thoughts, a site with the tag line "in pursuit of crafty mathematics," and I ended up really liking this design. My version has 8 squares in it, of course. I used more math than I'm usually aware of using when I sew, and it was really fun. In this post, I'll talk about some of the math I used. In the next post, I'll talk about the actual construction of the tablecloth.

The mathematics behind the square pattern is a curve of pursuit, at least as I understand it. I haven't done the computations myself, but I think that the "spiral" (it's piecewise straight, but the mind easily makes it into a curved spiral) traced out by the corners of the squares represents the path four dogs (or mice) would take if they were all mutually chasing each other at constant speed. Pretty cool, huh? The way I thought of constructing it was just rotating a smaller square until its corners hit the sides of the next larger square.

I decided that I wanted a constant ratio of sizes of squares, and I thought that it would be cool if the innermost square were parallel with the outermost. I did a simple high school geometry proof to make sure that the angle of successive squares was additive.

The left picture is a schematic of a few of the squares. The right is a blow-up of the way the angles in question are situated. The variable x represents the amount of rotation at each layer. The variable y represents the total rotation after two layers. I used the fact that the sum of angles of a Euclidean triangle is 180 degrees and that vertical angles are congruent to conclude that y=2x, which means that rotation is additive. (OK, technically I showed that the rotation would be additive if the rotation is by the same amount each time, but if x were replaced with z for a different amount of rotation, we would find that y=x+z.)

There might be an easier way to see that rotation would be additive in this case. It's obvious that it is additive if the vertices of both angles x are in the same place, so if the squares all had a corner in common. But I was worried that maybe something weird would happen when you slid the second angle along the side of the first triangle. This proof told me that nothing weird would happen. I have learned that my intuition is often wrong in math, so it was nice to have a proof so I knew for sure.

Since 13x7=91, which is close to 90, I decided that each square should be rotated by about 13 degrees to get the sides of the last square to be parallel with the first. I wanted the tablecloth to have a diagonal length of 60 inches because my vision was for it to be rotated and put on their table diagonally, and based on their table size, I thought 60 inches was a good length.

I couldn't actually find our calculator. (Update: I found it recently when I accidentally knocked it off the back of the work desk. But the batteries are dead anyway.) Instead, I used the google as a calculator to figure out what the side length of the largest square should be to give diagonal length 60. (Answer: 60/sqrt(2), about 42.4 inches, conveniently about the same as the width of the orange fabric I was using.)
 Then I set up an equation to figure out the ratio between successive square side lengths. In the picture to the left, I assume the total side length of the largest square is 1 and figure out where the second square touches the first using the equation I wrote at the bottom. To figure out the side length of the smaller square, I used the Pythagorean theorem with the x and 1-x to get the hypotenuse of that triangle, which is the side length of the second square. After getting a decimal approximation of that side length, I computed all the square side lengths to three significant figures. That should have been my first clue that I was greatly underestimating the difference between theory and practice: my tape measure goes down to 16ths of an inch, and my cutting is considerably less accurate than my tape measure.

Here is a sketch of my plan on poster board.

Even with the relatively good precision of pencil and paper, the innermost square didn't line up very well with the outermost square. But it looked cool anyway.

Unfettered by banal practicalities such as the realistic amount of precision I could hope for when cutting the fabric, I double-checked my numbers and went off to cut the fabric.

Tune in tomorrow (or whenever I get around to posting again) to read about the differences between theory and practice that I encountered during this project! (Compelling cliffhanger, or most compelling cliffhanger?)

Hyperbolic scarf

This is the first crochet project I attempted, and I've made it several times now. It's quite easy: chain a bunch, double crochet twice in each chain stitch, and then double crochet twice in each stitch on the next row. I like it because it's funky and mathematical. See, this is a great example of negative curvature, and as a hyperbolic geometer, I dig that. Here's a website where you can read a bit more about the mathematics of crochet. (The website is maintained by a math professor friend of mine who is using crochet to help get students excited about geometry. It's pretty cool! It also has some links to other websites about crochet and math.)

(Math digression in blue because I like blue.)

In case you're interested in the math, here's a little background. In 10th grade, you probably took a geometry class. In it, you learned about triangles and writing proofs using facts about how much total angle different shapes have in them and so on. Well, that was something called Euclidean geometry. In Euclidean geometry, the plane that you kind of base everything on is flat. One feature of Euclidean geometry is that parallel straight lines remain the same distance apart forever, and there is only one line parallel to a given line that goes through a given point. (This is called the parallel postulate.) Euclid thought that this was the only consistent geometry, but it turns out that there are consistent geometries in which your "plane" is not flat. In fact, all of them involve replacing the parallel postulate with a different axiom. One non-Euclidean geometry you are probably fairly familiar with is spherical geometry. Imagine that you are living on the surface of a sphere (not too hard, right?). That is called positive curvature. You can see the differences between positive and flat curvature most easily if you imagine a plane flight between points that are very far away from each other. The shortest distance between two points on the earth looks curved on a map. You'll also notice that any two of these curves, which are called great circles, will intersect. (For example, all the lines of longitude intersect in the poles.)

Well, hyperbolic geometry is the study of negative curvature. It's a bit harder to visualize than positive, but the way I think about it is that there is "extra surface area" for a given perimeter. "Parallel" (i.e. non-intersecting infinite) lines move away from each other. There are an infinite number of parallel lines between a line and a given point not on that line. One way to visualize a portion of the hyperbolic plane is to imagine a saddle. Some curves on it go up, and some go down. (In contrast, on a sphere, if you put two perpendicular lines on the north pole, they would both go down.) I find that sewing and crochet are really great media for displaying negative curvature because you can force your fabric to have "too much area" and pucker in a pretty way. That's what this scarf does by putting "too many" stitches into the second row (two stitches for every one stitch in the first row). The fabric has to twirl and buckle to make room for all those extra stitches. People who make hyperbolic crochet like to play with different increase ratios. 2:1 is a very steep increase. 6:5 is very pretty. To do that, you make six stitches in the second row for every five stitches in the first row. No matter what your increase ratio is, though, you will learn that exponential growth is very very fast!

Hyperbolic geometry is pretty neat. And you can be employed by it! People like me make a living studying spaces of hyperbolic surfaces.

OK, that was a long math digression. Back to the crochet. I like making this gift for people. Most recently, I sent a purple one to my MIL for her birthday. The green one above is mine, and the multicolored one I sent to one of my best friends for her birthday. It's easy enough that a crochet newbie like me can get the hang of it really quickly and whip one out in just a few hours.