Friday, November 23, 2012

The Math Behind Metamorph

On November 13, I published the pattern Metamorph.  This pattern was a long time in the making, and involved taking a journey deeper into the world of Topology than I had ever previously attempted.  This blog post is my attempt to document that journey.  Apologies in advance to individuals fluent in the language of Mathematics -- I only play a mathematician on YouTube.

The tale goes like this...

About 3 years ago while I was messing around with little form studies, I thought it would be cool to have a knitted form that looked like a torus (or donut) but had a whirlpool in the middle.  I knitted a tube, folded it, offset one edge, and grafted.  Here's what I got. 

Not exactly what I'd pictured, but interesting enough. I documented my experiment and considered it done.

Now I should mention: if there's something in my hands, I will probably play with it absent-mindedly.  So the next thing I knew, I looked down and saw this...

Given that this I’d just done a series of explorations dissecting the moebius form and thought I understood this form pretty well, this little object just about turned me on my head.

At its core, Metamorph is simply a torus. But when you offset one edge before sealing, this creates periodic harmonic orbits around the longitude, which introduces torsion -- or energy -- into the form. 

Image credit:


Image credit:

Mathematical models like the ones above show the periodic orbits.  With a knitted form, the energy created by these orbits manifests as folds in the fabric. 

After a little manipulation, the form will naturally relax into whatever shape requires the least amount of energy to maintain. In this case, this is the least-energy shape: 

Two conditions determine the least-energy shape:

(1) the amount you offset one edge before sealing, and
(2) the ratio of width:height. (Note that "width" here refers to the width of the knitted tube when laid flat, which is 1/2 the total circumference.) 
When the width:height ratio is 1:1, AND the amount of offset prior to sealing is 45 degrees, then after some gentle manipulation of the fabric, you will get a form with a single fold going all the way through the meridian. Or, a 90-degree offset will yield a form with two folds. So, Metamorph is divided into 8 equal segments because this affords a very simple mapping of n-button offset = n folds.

Well and good. But what if my width:height ratio isn't 1:1? Ah, I'm so glad you asked.  The conditions that correspond to 1 or 2 folds are continuous functions:

If height is less than 1/2 the width, or more than 2x the width, the form gets a little unruly.  The graph shows a comfortable range of sizes.
﻿﻿﻿Let's say you're following the Metamorph pattern and you suddenly run out of yarn.  Your width is 12", but your height is only 10".  If you divide your form into 8 equal segments as directed in the pattern, you will not get neat-and-tidy folds when you button your form together.  BUT, you can still get a form with nicely-defined folds IF you find your position on the graph above and alter your degree of offset accordingly. 

1.  Figure out the width:height ratio, given width = 1.
12:10 --> 1:0.83

2.  Pick a spot on the x-axis that looks like it corresponds to (1:0.83) and move up to see about where you land on the continuous functions, then left to the y-axis to see how many degrees of offset correspond to the point on each line. In this case it looks like somewhere around 36 degrees for a single fold, or 1/10 of the total circumference. 72 degrees will get you 2 folds.

So you can still have a mapping of n-button offset = n folds if you divide your tube into 10 segments instead of 8. 

Or you could also go the other way: say you have 12 buttons and you want to use them all on your Metamorph.  Divide 360 by 12 and you get 30; this time the matching width:height ratio for n-button offset = n folds would be about 1:0.67.  Or you could double it up: with a ratio of 1:1.33, then if you divide your form into 12 equal segments, a 2-button (60-degree) offset will give you 1 fold and a 4-button offset will give you 2 folds. 

Is your head hurting yet?  Mine is.

Now, adventurous souls may be wondering: what will my form look like if I go outside those lines?  With the 12-button, 1:1.33 example, what if you offset by some odd number of buttons?  Again -- I'm so glad you asked! 

What you get is something like this:


This is a snapshot of a 1:1 form with a fold that goes only part of the way through the meridian.  I don't remember exactly what I did, but based on how far down the fold goes (looks like about halfway to me), my guess is I gave it a shift of about 22.5 degrees, or half of what it would have taken to get a single fold across the meridian, based on the graph above. 

I owe a huge debt of gratitude to sarah-marie belcastro, Joshua Samsor, and Yonatan Munk for helping me get this deep into the wonderland of Topology. 

Other fun places that I discovered along this journey:
Wikipedia's page on the torus
Strange Loop by Morgen Dammerung
Plug-ins for modeling the twisted torus
The Twisted Torus and Knots by Jenny Buontempo
Sketches of Topology

Now, to find my way back up this rabbit-hole... :)



  1. I read this while knitting your double heelix socks, and am really inspired by your work! My brother who has a PhD in math was interested, too :) Thanks!