   The distinctive crenellated forms of the Hyperbolic Crochet Coral Reef are variations of a mathematical structure known as hyperbolic space. This aberrant geometry is an alternative to the canonical flat or Euclidean geometry that we learn about in school and the spherical geometry that represents the surface of our earth. Mathematicians spent hundreds of years trying to prove that anything like hyperbolic space was impossible only to discover in the nineteenth century that the laws of mathematics necessitated this form.

In 1997 Dr Daina Taimina discovered that it was possible to model hyperbolic space using crochet, an innovation that surprised the mathematical world. Her geometrically precise models are now used to help introduce university students to the subject of non-Euclidean geometry, the mathematics that underlies general relativity and which will therefore help us to understand the structure of our universe.

Mathematically, the three geometries may be understood by the behaviour of ‘parallel’ lines. On a Euclidean surface parallel lines stay the same distance apart forever. On a spherical surface all parallels intersect – think about the lines of longitude meeting at the poles of our earth. On a hyperbolic surface, parallel lines diverge. Where spherical space has positive curvature and flat space has zero curvature, hyperbolic space has negative curvature – it is the geometric equivalent of a negative number. The unique properties of this geometry were a source of fascination to the Dutch artist M.C. Escher, whose Circle Limit series of woodprints explore the rich tessellations possible within this space.

Nature also delights in the hyperbolic form but does not feel compelled to realize geometrical perfection. Just as there are many things in nature that are spheroid, such as eggs, but no such thing as a perfect natural sphere, so too the organic world abounds in not-quite-pure hyperbolic structures. The Hyperbolic Crochet Reef Project takes its cue from this domain of organic ‘imperfection’.