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.