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.

Set theory and knitting notation: update

Previously on Arachne's Loom....
I got stuck when this question came up: Can a cyclic group have repeating elements, such as (1, 2, 2, 1) or equivalently, (1, 1, 2, 2)?
See, in the set theory model of knitting I'm working on, there are only two elements: knit and purl. Obviously, those repeat a lot in knitting patterns. So the pattern/cycle of stitches in a given position in the row is going to have repetitions.

I emailed my professor, and he said he'll "have to give it some thought." Crazy!

Also, I have to clean up some mistakes from last night.

Consider the 2x3 rib pattern over 20 stitches.
Knitting notation:
Row 1: K2, P3, repeat to end of row.
Row 2: K3, P2, repeat to end of row.

In set theory terms, I notated the domain (Row 1) and range (Row 2) as follows:
Row 1: KKPPPKKPPPKKPPPKKPPP
Row 2: PPKKKPPKKKPPKKKPPKKK

MAJOR IMPORTANT NOTE: Row 1 is written left to right; Row 2 is written right to left.

That's because in knitting, you turn the work at the end of each row. Alternating rows having you facing the "right side" (outside) or "wrong side" (inside) of the garment. That's just the mechanical nature of knitting.

So your progress goes like this:

--> -->  --> --> --> -->     End of row!
<-- <-- <-- <-- <-- <--

[Clearly I'm a legit graphic designer. You get the idea.]

What do knit and purl mean, anyway? A stitch is just a loop of yarn pulled through the loop in the previous row. The loop faces you. Knit: the new loop is pulled through from back to front. Purl: from front to back.

So it follows that purling from the back (wrong side) is the equivalent of knitting from the front (right side).
That's why a rib pattern, which I notated

Row 1: KKPPPKKPPPKKPPPKKPPP
Row 2: PPKKKPPKKKPPKKKPPKKK

Looks like this:

 

The neat little rows are the result of knitting 2 stitches, then purling 2 right below them, and so on. Stitches you knit on a given side will stand OUT on that side; stitches you purled will recede. Flip it over and you have the reverse. (Knitting exploits this property to get all kinds of different textures.)

So if you purl on the even rows (facing the wrong side), the resulting stitches look like knitting on the right side.
So in the set-theory model, it might make more sense to represent the stitches AS THEY APPEAR from the right side.

The ribbing pattern becomes:
Row 1:  KKPPPKKPPPKKPPPKKPPP
Row 2:  KKPPPKKPPPKKPPPKKPPP

Looks more like the picture, doesn't it? Each rib looks like it's made of the same kind of stitch.

Ribbing is usually reversible, as long as each rib is the same width; for example, K3, P3, etc. That's called a 3x3 rib, as opposed to the 2x3 rib I used as an example.

Let's look at some-reversible patterns to confirm that this all-left-to-right notation makes more sense.

DOUBLE SEED STITCH (again), 10 stitches per row
Knitting notation:
Row 1: K1, P1, repeat to end of row.
Row 2: "        "
Row 3: P1, K1, repeat to end of row.
Row 4: "       "

Set theory notation:
Row 1: KPKPKPKPKP
Row 2: KPKPKPKPKP
Row 3: PKPKPKPKPK
Row 4: PKPKPKPKPK

Okay, that's just making it more confusing because the two coincide. Next.


DIAMOND BROCADE
17 stitches per row (can be any 8x + 1), repeat every 8 rows.

Knitting notation:
Row 1: K4, [P1, K7], P1, K4
Row 2: P3, [K1, P1, K1, P5], K1, P1, K1, P3
Row 3: K2, [P1, K3, P1, K3], P1, K3, P1, K2
Row 4: P1, [K1, P5, K1, P1], K1, P5, K1, P1
Row 5: [P1, K7], P1, K7, P1
Row 6: P1, [K1, P5, K1, P1], K1, P5, K1, P1
Row 7: K2, [P1, K3, P1, K3], P1, K3, P1, K2
Row 8: P3, [K1, P1, K1, P5], K1, P1, K1, P3
(The segments in brackets can be repeated to make the work wider.)

Set theory notation:
Row 1: KKKKPKKKKKKKPKKKK
Row 2: KKKPKPKKKKKPKPKKK
Row 3: KKPKKKPKKKPKKKPKK
Row 4: KPKKKKKPKPKKKKKPK
Row 5: PKKKKKKKPKKKKKKKP
Row 6: KPKKKKKPKPKKKKKPK
Row 7: KKPKKKPKKKPKKKPKK
Row 8: KKKPKPKKKKKPKPKKK

There's a different cycle for each column (position in the row).
Column 1 goes (1, 1, 1, 1, 2, 1, 1, 1). So do columns 9, 17, etc.
Column  2 (or any column 8x + 2): (1, 1, 1, 2, 1, 2, 1, 1, ).
Column 3: (1, 1, 2, 1, 1, 1, 2, 1).
Column 4: (1, 2, 1, 1, 1, 1, 1, 2).
Column 5: (2, 1, 1, 1, 1, 1, 1, 1)
Column 6: same as Column 4
Column 7: same as Column 3
Column 8: same as Column 2

....

ZOMG! It's a cycle of cycles!







Not just that, but the columns and rows repeat in the same pattern.
Symmetry jackpot.



To be continued.

Set theory and knitting notation, cont.

Still working out the details, but seriously, rows of stitches are like sets, and knitting patterns are like groups or functions that act on the sets! It's just like transformation theory in music!

Say you have a tone row, where the notes C through B are represented as pitch classes 0 to 11.

Random tone row: 3 7 6 2 1 9 5 8 11 10 4 0

Function: T3 (transpose up 3 chromatic steps).

Elements in the domain (original tone row) map to elements in the range (resulting tone row) according to the function x + 3 (add 3).

Resulting tone row: 6 10 9 5 4 0 8 11 2 1 7 3

In other words, 3 maps to 6, 7 maps to 10, etc.
You could write this as three cycles: (0, 3, 6, 9), (1,4, 7, 10), (2, 5, 8, 11)
These are called orbits: an element can't map to anything outside its orbit, no matter how many times you apply the function.

Anyhoo, the function turns one tone row into another that is a rearrangement of the same elements. Similarly, in knitting, any function (pattern) will take a row of knits and purls and produce another row of knits and purls in some other order. And that's how you knit cool textiles. Just two different stitches, but mix them up and the possibilities are endless!

Off to geek out. I mean, sleep.

Set theory and knitting notation

Why am I even still up? Once again, the knitting muse is luring me away from constructive pursuits like practicing or sleeping. I don't even know if this idea is unique, or if it's been done before, but I just came up with a mathematical way of notating knitting patterns: mapping.

It's simple, really. Basics of set theory: elements in the domain (first set) map to elements in the range (second set) according to whatever function you're using. Example:

Domain:     1   2   3   4
Range:       2   3   4   1

In this mapping, 1 "goes to" 2, 2 goes to 3, 3 goes to 4, and 4 goes back to 1.
If you start with the ordered set [1, 2, 3, 4], after one iteration of the function, you'll have [2, 3, 4, 1].  After two iterations, [3, 4, 1, 2]; after three, [4, 1, 2, 3], and four iterations bring us back to [1, 2, 3, 4].
This can also be expressed as the cyclic group (1, 2, 3, 4). This notation illustrates the path of an element. It goes through the numbers in ascending order, as opposed to, for instance, (3, 1, 2, 4). In that case, 3 maps to 1, which maps to 2, which maps to 4, which maps back to 1.
I'd like to add at this point that I have no idea how coherent I am right now. It's late. 

In the case of knitting, the function is the pattern. It determines what stitch (knit or purl) you do based on the stitch in the previous row, in the same position in that row.

Here comes the leap! Let's notate a knit 1, and a purl 2.

A standard pattern will tell you the order of stitches in each row. For example, look at a 3x3 rib pattern worked over 24 stitches. The pattern will say something to the effect of:
Row 1: K3, P3, repeat to end of row.
Repeat.

Or to generalize (because I feel like that might be useful later), let's talk about an n by m rib pattern worked over x stitches, where x = (n + m) times some constant.
The abstract pattern will read:
Row 1: K n, P m, repeat to end of row.
Row 2: K m, P n, repeate to end of row.
Repeat.

Suppose n = 2, m = 3, and x = 20. That is, knit 2, purl 3, repeat over 20 stitches.
Row 1 can be written: KKPPPKKPPPKKPPPKKPPP
Row 2 "       " :           PPKKKPPKKKPPKKKPPKKK

Looks like a domain and range to me!

This rib is a 2-row repeating pattern, and every stitch is the opposite of the stitch in the same position in the previous row. It's pretty easy to express mathematically: K maps to P, and P maps to K. Or in the set theory model I'm making up right now, 1 maps to 2, and 2 maps to 1.

Ladies and gentlemen, we have a cyclic group!
(1, 2)

Yay.

Okay, let's try fancier knitting patterns.

DOUBLE SEED STITCH
This is a 4-row pattern worked over an even number of stitches (meaning the number of stitches in each row is even). Pattern reads:
Row 1: K1, P1, repeat to end of row.
Row 2: Repeat.
Row 3: P1, K1, repeat to end of row.
Row 4: Repeat.

This produces a cool texture because it's basically a 1x1 rib that's reversed every two rows.

In the set theory model, assuming x = 10 (meaning we're knitting something 10 stitches wide, like a headband):
Row 1: KPKPKPKPKP
Row 2: PKPKPKPKPK
Row 3: PKPKPKPKPK
Row 4: KPKPKPKPKP

The cyclic group would be written (1, 2, 2, 1).
Wait, is that even a legitimate group? Excuse me while I consult my Intro to Abstract Algebra book. Groups... Cyclic Groups... here we are.
"If the multiplicative group G contains an element a such that G = {a^k, where k is a positive integer}, we say that G is a cyclic group and that G is generated by a or that a is a generator of G." (McCoy & Janusz, An Introduction to Abstract Algebra, 2001)

And this, kids, is why you take an actual course, instead of winging it like I'm doing now.

To be continued.

Bodies and geometry

Awesome, broad-ranging educational blog:
 http://longstreet.typepad.com/thesciencebookstore/2010/09/leggy-human-geometry-and-perspective-in-busby-berkeley-musicals-image-dump.html

I happened upon it when Googling Busby Berkeley, the master of geometric choreography.



Working in the 20s, 30s and 40s, Berkeley choreographed for Broadway and Hollywood. His dance numbers involved dozens or hundreds of showgirls in elaborate, shifting geometric patterns.
Though innovative in his stage and film techniques, Berkeley was accused of collectivism - for reducing the dancers to cogs in the choreographic wheel (see above). But if you ask me, that's pretty much the fate of chorus girls.

Just consider A Chorus Line, the award-winning 1975 musical choreographed by Michael Bennett. This scene addresses the desperation of dozens of dancers competing for spots in the chorus. Here are the lyrics: http://www.theatredance.com/aclly1.html