Holding Theorems in Our Hands

For years, hyperbolic space was purely theoretical. It was emerged out of a failure to prove one of Euclid's axioms: which states that there is only one infinate straight line that can run next to another without intersecting it - a parallel line. Mathematicians eventually proved that there were many lines, running in many directions, that would not intersect the original, but they existed in hyperbolic space.

All well and good, but when you start getting into theoretical spaces you kind of lose me. Mathematicians such as Poincare and artists like Escher have given us abstract representations of hyperbolic space. Poincare gave us the hyperbolic disk - an abstract representation of tiled shapes (such as heptagons) in hyperbolic space - necessary because the sum of their angles exceed 360 degrees. Poincare achieved this by reducing scale exponentially. Escher even put fish in his, always a plus on this blog.

They're quite gorgeous, as well as useful. I love repetition as motif, and find myself agreeing with the Institute of Figuring online exhibition when it notes that, "If, as the Moors believed, repeated patterns connote the divine, we might conclude that Heaven itself would be a hyperbolic space."

As lovely as they are, however, they're just analogies. Enter Daina Taimina, a mathematician who learned the dark arts of knitting and crochet as a child in Latvia. Her crochet models are a mathematically accurate example of hyperbolic shapes. They are not just representations, they are not abstract. They are, as she says, theorems we can hold in our hands. Check out the Institute for Figuring's online exhibition to see models that refute Euclid's parallel postulate, and buy her book here.

This is a model of a cone in hyperbolic space:

And this is where the marvelous ocean comes in. Hyperbolic-esque shapes are found in nature too - lettuce leaves, for example, or the frills of a nudibranch. And, of course, the coral reefs. From this realisation came the Hyperbolic Coral Reef, a global craft project that aims to draw attention to the frightening and rapid destruction of our coral reefs.

I saw examples of the Sydney HCR on display at last year's This Is Not Art festival in Newcastle. They are surprising and beautiful - colourful and intricate organic shapes, painstakingly crafted by hundreds of dedicated people.

Hyperbolic Reef projects exist all over the world. As beautiful as they are, we must remember they are but imitations - analogies, abstracts - of the real thing. The real thing, sadly, we are losing too soon.