Hyperbolic Space

Poincaré disc model of hyperbolic space from fractalsciencekit.com

Margaret Wertheim gave this interesting lecture called "The Beautiful Math of Coral".

  

She does a thorough job of explaining the math of hyperbolic space here on her website, the Institute for Figuring. There are photos of the gorgeous crocheted coral reefs, too.

In her words...

The Crochet Reef Project was inspired by the technique of hyperbolic crochet originally developed by Dr Daina Taimina, a mathematician at Cornell. In 1997 Dr Taimina discovered how to make models of the geometry known as "hyperbolic space" using the art of crochet. Until that time many mathematicians believed it was impossible to construct physical models of hyperbolic forms; yet nature had been doing just that for hundreds of millions of years. It turns out that many marine organisms embody hyperbolic geometry in their anatomies - among them kelps, corals, sponges, sea slugs and nudibranchs. Thus the Crochet Reef not only looks like a coral reef, it draws on the same underlying geometry endemic in the oceanic realm.

There are very good reasons why marine organisms take on hyperbolic forms: this geometry is a marvelous way to maximize surface area in a limited volume, thereby providing greater opportunity for filter feeding by stationary organisms.

An unidentified folded coral in Flynn Reef, part of the Great Barrier Reef, near Cairns, Queensland, Australia. By Toby Hudson.

Crochet works because it is an easy way to increase stitches in each row to produce ruffling. I haven’t crocheted since childhood, but I do remember increasing stitches when I wasn’t supposed to. Maybe I should try a coral reef.