Thoughts on knitting and calculus

Solving calculus problems, like knitting, is a meticulous process, and the smallest careless error will distort all subsequent work. 

Say you drop a stitch. Not only will your knitting unravel, but subsequent rows will have the incorrect number of stitches. That's a very simple example. Something like turning the heel on a sock requires greater mathematical precision, and miscounting the number of stitches at any point in the pattern will lead to a very distorted sock. 

Say you're finding the differential of a function of two or more variables, like pressure of a gas as a function of temperature and volume: P = 8.31T/V. First you have to take the derivative of P with respect to T. Then you take the derivative of P with respect to V. Then you find dV (the change in V) and dT (change in T) based on the values you're given. Plug all those values into the formula for the differential. Mess up at any point (or omit a negative sign, like I did), and your answer will be wrong. 

Educational application: does knitting create mathematical minds? 

Generalized crochet Klein bottle instructions

Basic idea of how to crochet a Klein bottle. Klein bottles can have many different proportions, of course; you can mess around with the curvature.

Close-up of the Tonnetz

Let's take a closer look at the Tonnetz and its properties. First off, the triangles defined by three adjacent points represent major and minor chords. As you can see, isometries (geometric transformations) in the Tonnetz correspond to modulations (changes to a nearby key) in music.

Again, the Tonnetz can be tesselated/tiled indefinitely to fill a plane. Its sides connect with the sides of other Tonnetze. So picture the keys going on forever, as they do in music theory.

Baby's First Tonnetz 2.0

You know how I said you could model the Tonnetz (all major keys) on the surface of torus?

Each rectangual region is a Tonnetz, representing all twelve major keys separated by perfect fifths. There are twelve tesselations in the diagram. See how the top connects with the bottom, and the two sides connect? Voila, torus!

Additionally, I mapped the Circle of Fifths to the color wheel: keys that are closely related (close together on the Circle of Fifths) are closer on the color wheel.

I'm making a crochet model of this now. It will be soft and fuzzy and colorful, hence Baby's First Tonnetz.

By the time I finish this project, I will permanently associate the keys with twelve different colors. C will always be red. Give it to a baby and the kid will probably grow up that way too!

I call it "Baby's First Tonnetz."

You start out by crocheting a torus (pictured: halfway through).

Crocheting completed, before stitching the edges together.

Stitched and stuffed!

Candid shot of my room at 3:00 am.

Then I added some embroidery floss in 12 colors, representing the 12 keys in the Circle of Fifths: C, G, D, A, E, B, F#, C#, G#(A flat), E flat, B flat, F, and back to C! Like the Circle of Fifths, the color wheel is also cyclic!

And voila! This is the other side. Continuous thread (12 strands linked toghether) in a continuous spiral around the torus.

Next up: making a larger torus that will make the Tonnetz easier to embroider. I tried to do just one iteration of the Tonnetz on the pink torus, but it became so distorted that the rows didn't line up into columns. Hopefully doing more than one tesselation will make it easier. I'll probably just embroider little discrete knots or points rather than a continuous thread.

Ruth's adventures in topological knitting and crochet

Below are photos of my foray into topological textiles over the last few weeks.

For those of yall just tuning in, topology is the study of curved surfaces.



1. My first attempt at knittnig a Klein bottle, which is a surface with only one side. In textiles (knitting, crochet, weaving), this means that the two sides of the resulting fabric are continuous.

If that didn't make sense, let's take a step back into two dimensions.
Consider a Mobius strip (pardon the missing umlaut).


You've probably made one in school by twisting a strip of paper. The original paper had two sides and two
pairs of parallel edges; the resulting Mobius strip has one side, because now the two sides are continuous, and only one edge.

To make a Klein bottle, you start with a very stretchy cylinder, turn one end inside out and then connect the two ends. This can only truly be done in four dimensions. In three dimensions, we can approximate it with a surface that has a self-intersection (where the surface passes through itself).
Nifty diagram here: http://inperc.com/wiki/index.php?title=Klein_bottle

And like the Mobius strip, this surface now has only one side! Not just that, but it's a closed surface (like a sphere), yet the interior is continuous with the exterior! Think of a beach ball (sphere). If you're on the inside, you can't get out, and vice versa. A Klein bottle is also a closed surface, but if you're inside, you can get out through the neck-like part. All thanks to the self-intersection, which, again, is how we cheat in three dimensions.


image from http://mediawingnuts.blogspot.com/2010/05/klein-bottle-or-klein-jar-either-way.html

It's hard to get your head around unless you have a physical model you can touch and turn around and around (because it's non-orientable) and stick your fingers into. This is where knitting comes in!

Here's the knitted Klein bottle from above, er, from the side with the opening.

Knitters will notice that I cheated by switching from stockinette to reverse stockinette at the point where the work turns inside out. That's one the problems with knitting; it doesn't always look the same on both sides, and a Klein bottle should, because the two sides connect.

My first Klein sock. Most socks have an inside and and outside; this one, of course, has only one side. I started with a sock pattern, left a hole in the top of the foot, made the toe extra long and narrow, drew it through the hole (had to cut the yarn and reconnect it), widened it to the width of the cuff, and connected the two ends.

On to crochet! Not only is crochet naturally reversible (looks the same on both sides), but it's sturdier and holds its shape better. Pictured here is a Klein change purse from the... top? You could call any side the top.

From the side (relative to the the top I defined).

The opening (in the bottom) where the money goes in.

The coins go in through the opening, through the neck, and fall back to the bottom. You have to rotate the shake the purse and kind of manipulate the coins through. The upside is it would really confuse a thief, and it makes you think twice about spending your money!

Next up:

The sleeve and shoulder of a life-sized shrug modeled on the tiny one I posted about earlier.

It looked fine from the front, but the back was all bunched up because the curvature of the two circular pieces was greater than the curvature of my back. Turns out my super-easy shrug pattern was too good to be true. Knitting and crocheting clothes would be a lot easier if the human body were more mathematical.  

Finally, the beginning of my pink torus, which I'm planning to embroider the Tonnetz on.

Again, the project had a surprise in store: as you can see, the torus is beginning to twist into a hollow corkscrew shape. Which is kind of cool mathematically - it's like the equivalent of a computer error message - but it's not what I had in mind.

I'm pretty sure this is due to the nature of chrochet: it's not a series of rings stacked on top of each other, but each successive row is shifted over half a stitch, because you make a stictch between two stitches in the previous row.

I'm hoping to work around this by making several shorter segments and sewing them together into a torus. If my calculations are correct (okay, more like a nonmathematical hunch), the curvature of two mirror-image segments should cancel each other out, giving me a nice symmetrical, non-squiggly torus. Here goes! 

Well, I tried.

First attempt at a cross-stitch interpretation of Escher No. 96, System IV-D:

The original is inset. The black points in my sketch (graph, really, if we can speak of graphing art!) correspond to the red points in the original. I did the brown (left-facing) swans in brown and the white (right-facing) ones in green. In group theory and music theory, we call these two cycles, or orbits.
It didn't work for a couple of reasons which I'll tweak the next time around. Generally, I wasn't as meticulous as Escher had to be to make this work (to make it obtain, as they say in abstract algebra). Then, I didn't realize the points at the top of the swans' heads should be the exact same distance from the point at the back of the swan's head (corresponding to the nearest corner of a square in my graph) in both directions, because it somehow escaped me that the right- and left-facing swans are identical, just reflected. D'oh.
The other problem is that I only made the squares ten by ten stitches, represented by the smallest units in the graph. This just made it harder to imitate Escher's original swan shapes. It also made the shapes more pixelated, of course. Since Escher's tile shapes are only vaguely realistic to begin with, I need to be as precise as possible to preserve the shapes. Next time I'll do at least twelve stitches to a side, though that will take a lot longer and probably make my eyesight deteriorate faster.

The good news is that we have an equivalent of this is music theory, which is what I have to write a paper on, haha. Basically, the points in the graph (at the intersections of lines) are equated to points in the Tonnetz, which each represent major and minor triads.
The better news is that in knitting, we have the additional property that knitting is the "inverse" of purling, which also has a parallel music: major and minor triads. Invert a major, you get a minor - a very different sound.

So if I do a visually engaging representation of the Tonnetz in the knitting medium, it would look kind of like one of Escher's "tiled" works, consisting of two interlocking cycles of shapes, one right-facing and one left-facing. ("Cycles" in the group theory sense come up a lot in algebraic knitting.) One cycle would be in knits (stockinette stitch), and the other would be in purls (reverse stockinette). Flip it over and you see the same pattern.

This idea of reversible fabrics HAS PRECEDENT IN KNITTING!



This knit is reversible; flip it over and it looks the same. It's also a simple example of a tile pattern:

As with Escher, the pattern repeats both horizontally and vertically.

This is also a property of the Tonnetz ("network of tonal relations"), a two-dimensional model of all the major and minor chords in the diatonic system.



This is actually a bad picture of the Tonnetz because it doesn't reflect enharmonic equivalence: for instance, C sharp equals D flat, C double sharp equals D double flat, etc. The concise Tonnetz is much smaller, without all that redundancy.
See how the patterns (cycles) or major and minor triads resemble the right- and left-facing swans? The major triads always point up, and the minor ones always point down. As in Escher, the two cycles are reflections of each other.

Wanna know something else cool?




This blog can do animations!
No, besides that. You're looking at the 3-D Tonnetz. Since it repeats both horizontally and vertically, it can be  represented on the surface of a torus (donut).

You can also knit tori.

To be continued.