Hyperbolic Crochet

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Read about the intersection of mathematics and the handcraft of crochet.

Three straight lines- geodesics

The Auckland museum’s Seagardens Project was on display in the Oceans Gallery until May 16 2010 showing crocheted models of hyperbolic sea life.

The Seagardens Project is affiliated with the Hyperbolic Crochet Coral Reef project, initiated by two Queensland sisters in homage to the threatened Great Barrier Reef. It was based on the first easily usable physical models of hyperbolic space, developed by mathematician Daina Taimina in 1997, using ideas from William Thurston.

Illustrates the narrow angles of a triangle on a hyperbolic plane

Taimina, a skilled knitter and crocheter, realised that crochet was the best medium to demonstrate the non-Euclidian properties of hyperbolic geometry. In this geometry, for any given line L and point p not on L, there are at least two distinct lines through p that do not intersect L.The angles of a triangle in hyperbolic space also add up to less than 180°.
In March Taimina’s book, Crocheting Adventures with Hyperbolic Planes, won the 32nd annual Diagram Prize for the oddest
A hyperbolic soccer  ball model
book title award of 2009, from a record 90 entries. The book and the projects reflect a unique combination of maths, community art and marine biology.

The Seagardens and HCCR projects require contributors to use a simple algorithm to construct their planes. Former Seagardens Co-ordinator Glenys Stace said this is easy for complete beginners. She suggested crocheters choose any number of stitches (around five works well), crochet that number and add another stitch in the last hole. Repeating that process quickly develops ruffling that can be wrapped into a huge variety of shapes, each called an embedding in 3-space.

Unlike a flat sheet of Euclidian paper, which can be wrapped only into a simple cylinder or a cone, hyperbolic planes can be wrapped into multiple flutes without deforming their geometry. In the Seagardens installation, these simulate anemonies, branched corals and loopy kelp.

Christine and Margaret Wertheim co-direct the Institute For Figuring, an educational organisation based in Los Angeles.Their HCCR project has developed six different crochet coral reef exhibitions.

Post your creation to Vincent Lipanovich, Auckland Museum, Private Bag 92018, Auckland, phone him on 09 309 0443 x 924, or email vlipanovich@aucklandmuseum.com

Book cover
Blue:Three straight lines - geodesics
- in a hyperbolic plane pass through a point external to the line at the bottom without intersecting it.
Red: Illustrates the narrow angles

Black and white:A hyperbolic soccer ball model, created by Taimina’s son Keith Henderson, replaces the five-sided pentagons with seven-sided heptagons and includes seven six-sided hexagons for every five in the original shape. Instead of closing into a sphere, the surface opens out and curves away from itself.

See also
Crochet coral reef site

Daina Taimina website

Institute for Figuring

Video: Margaret Wertheim TED talk