To begin with, you need to construct an appropriate "eutactic star" in hyperbolic space --- basically, a projection matrix which lines up higher-space lattice points to project discretely onto lower-space. Such objects are pretty thin on the ground: see Coxeter "Regular Polytopes". But that was the easy part: now you have to select which elements of the higher polytopes to retain and which to discard from your projection. There's no road-map for this stage. I recently spent several months attacking the generalisation of Penrose tilings to (Euclidean) 3-space. I eventually formed the opinion that --- despite the considerable literature on the topic, mostly by chemists --- no such object has yet been conclusively (or even convincingly) shown to exist. Like everybody else who seems to have thought about the matter, I'm sure it's out there somewhere --- but I couldn't hack the so-and-so either! Fred Lunnon On 2/21/11, Mike Stay <metaweta@gmail.com> wrote:
Has anyone worked out aperiodic tiles for the hyperbolic plane? Is there something analogous to Penrose tiles? Penrose tiles are a 2-d projection of a 5-d hypercube lattice; in 3d, there's a "cubic" tiling of hyperbolic space with dodecahedra where eight dodecahedra meet around a central vertex like cubes do in Euclidean space: http://reperiendi.files.wordpress.com/2011/01/dodecahedral.jpg If we took some "hypercubic" tiling of 5d hyperbolic space and projected down to 2d hyperbolic space, what would we get? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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