Hyperbolic Space

Though easily modelled in crochet and knitting as a steadily increasing ruffle, hyperbolic space was a radical discovery that challenged the old Euclidean notions of linearity.  In hyperbolic space, multiple "straight" lines may pass through a given point without intersecting, a feat not possible in Euclidean geometry and with fascinating implications for the understanding of the structure of our universe.

In biology, examples of hyperbolic planes abound, including ruffled lettuces and, in the ocean reef environment, sea slugs, nudibranches and flatworms.  This pattern of growth is a very efficient way to create maximal surface area in a limited space.

My own Orchid, crocheted in fine silver:

Further reading on hyperbolic space:

From the Institute for Figuring: a very readable introduction to the subject, with lots of illustrations and interesting links:   http://theiff.org/oexhibits/oe1.html  Particularly intriguing is their gallery of crochet models, especially the coral reef project.

The application of hyperbolic symmetry to tiling patterns, particularly favored by the famous Dutch artist, M.C. Escher:  http://www.sciencenews.org/articles/20001223/bob8.asp

Hyperbolic tile patterns by Jos Leys:  http://www.josleys.com/show_gallery.php?galid=262

A discussion of the influence of non-Euclidean geometry on modern art: http://math.mohawkcollege.ca/ocma/conf05/Post_Conf05/john_Modern_%20Art.pdf