Saturday, 21 March 2009

'Hyperbolic Planar Tesselations', by Don Hatch

John Baez's "This week's finds in Mathematical Physics" page often has links to math goodies. I haven't visited it for a while (where "a while" is probably measured in years), but I had a peek today, and it had a link to a site containing a whole collection of these beasties:

thumbnailof images from 'Hyperbolic Planar Tesselations' at, by Don HatchIt's a page by Don Hatch called Hyperbolic Planar Tesselations, and it's full of links to larger versions of the pretty pictures. The image selected on the Baez page is especially nice, because it shows the tiling that you can achieve in negatively-curved space by replacing the usual flat-spacetime hexagonal tiling with heptagons. These regular tilings don't work in a flat plane. If we extrude a flat plane in one direction, then the amount of space per unit area, as judged within the plane, is less than we'd expect. If we extrude in two opposing directions (to produce a "saddle" or "pringle" shape), then as we draw larger shapes on the surface, they include progressively more area that we'd normally expect, thanks to all the folds and crinkles, and the resulting hyperbolic plane allows things like regular heptagonal tiling.

Okay, so I'm probably a sucker for tables of blue, black, and white geometrical figures, but even so, the "Don Hatch" page is really very nice. Some of the figures are reminiscent of Apollonian Net diagrams, which I'm quite fond of as fractal tiling systems, and which also in turn tend to correspond to maps of fractal-faceted solids with an infinite number of circular faces that you can achieve by continually grinding maximally-sized flat circular facets into the remaining curved surface of a truncated sphere:
Infinitely-truncated sphere, giving an infinite-sided polygon with circular faces, whose map corresponds to an Appollonian gasket
I put a quick illustrative connection map of heptagonal space on p.27 of the book ("3: Curved Space and Time"), but it was really just a crude sketch. So while my first reaction to the Hatch page was "Wow! Cool!", my second was, "Damn, I wish I'd done that".

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