Friday, 23 December 2016

Hyperbolic Crochet and the Pseudosphere

Warning - the author takes no responsibility for any sleep deprivation that may occur as a consequence of trying this at home.

I have a friend on Ravelry to thank for my hyperbolic journey into exponentially expanding surfaces. She posted a link to Gary Hayton's knitted felted Fibonacci Pseudospheres, which just beg you to imagine, wow, however did he make those? The title sent me off to rediscover the Fibonacci numbers and spend time pondering how they might be incorporated into a circular knitting pattern. 
Just as I was developing terminal brain ache, the same good friend explained that all you have to do to create a hyperbolic shape is to make increases at a constant rate. Starting with a magic loop of six double crochet stitches (single crochet in America), I began crocheting two stitches of heavy, inelastic tapestry wool yarn into every stitch, spiralling endlessly outwards.
The initially flat surface develops frills that wrinkle up together in three dimensions, mine formed a pseudosphere after about eight rounds. Watch this video and you'll see what I mean. Because I made two stitches into every one, the stitch count doubled with each round 6 - 12 - 24 - 48 - 96 - 192 - 384 - 768 - 1,526 - 3,072, so deciding to do the final round with crocheted picot edging in a colour change yarn took rather more time and yarn than I had bargained for. Running low toward the end, I finished the last stretch with a plain crochet edge.
Himself was pleased to refer to my new pet as 'The Golden Brain'. A bit more googling about led me through whole coral reefs created with crochet to the brilliant mathematician who made the leap from abstract hyperbolic geometry into tangible yarn constructions. In this lecture, she tells her story so simply, even I felt I understood negative curvature - until I tried to explain it to himself.

Much as I love my pet brain, I had no answer when asked what it was for. Daina made hers to teach students, the crocheted undersea world was intended to raise awareness of coral death due to pollution. Trying to dream up some function to justify my urge to go hyperbolic again, I crocheted a basic hat shape in coloured Romney handspun wool yarn.
Working two stitches into every one for a few rounds in a darker shade of the same wool created a frilly brim. All might have been well had I left it at that, only I carried on, using a wild skein of coreless corespun 'art yarn', only finishing at 2am, when I had used up the last of the wool.
Daina Taimina says in her lecture that Wolfgang Bolyai warned his son of the perils of pursuing non - Euclidean hyperbolic mathematics. 
The same might be said of hyperbolic crochet.
'For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.' 
Come on in, the water's lovely.


  1. Love the hats! May have to try this myself once the holiday madness stabilizes next year.

    A small quibble - Fibonacci sequence follows as addition of 2 previous numbers to create the next one: 1, 1, 2, 3, 5, 8, 13, 21, 34 and on into infinity. The next fascinating piece of the puzzle - if you divide the larger of a pair of numbers by the smaller the result is 1.618. Divide the smaller by the larger and the result is .618 (Accuracy increases the higher you go in the sequence.)1.618 is a constant named phi. It is the proportions of the golden rectangle, displayed in the Parthenon.

    I'll stop here, as Fibonacci, phi, the patterns of nature are a passion of mine.

    Thank you for your passion of knitting and dyeing and sharing with the rest of us.

    1. That is fascinating. Now, however can 1.618 be incorporated into knitting or crochet ....

    2. Fibonacci sequence number pairs are found in sunflower heads (and all daisy family flowers, different left- and right-hand spirals in different flowers), pine cones etc. Chambered nautilus have increasing phi rectangles nicely displayed in shell section. The Victorians loved this sort of thing.

      My sister once knit a cowl for me in two colors of yarn using Fibonacci sequence for color changes.

  2. Wow, madness! This would be an excellent bath scrubbie or attached to a stick abottle cleaner! Happy holidays / merry Xmas / have a great day everyone!

    1. The Golden Brain cannot be expected to do mere skin and bottle cleansing - I am relying on it only for abstract brilliance. Happy New Year :)

  3. I had found the hyperbolic crochet 'things' a couple of years ago when I was still teaching geometry. I toyed with making one for class use, but retired before I actually did it. It makes a good 'wreath' edge for a hat. I can see it for renaissance fair wreaths too. Beautiful. Helen

    1. I think the hat does have a Pre-Raphaelite vibe. Doesn't fit very securely though - shake my head, the thing slips sideways and the look goes seriously Post-Bacchanalian