Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!wuarchive!zaphod.mps.ohio-state.edu!uakari.primate.wisc.edu!aplcen!haven!umd5!newton.cs.jhu.edu!callahan From: callahan@cs.jhu.edu (Paul Callahan) Newsgroups: comp.theory.cell-automata Subject: Re: Spherical CA Message-ID: Date: 5 Oct 90 17:13:36 GMT References: <90277.175712HFIHC@CUNYVM.BITNET> Lines: 48 HFIHC@CUNYVM writes: >Has anyone ever designed and/or implemented a CA for a spherical surface? >If so, how do you tile a spherical surface so that no cell is larger, or >differently shaped than another? >Any ideas? A couple of people have pointed out the limited number of regular polyhedra available for this purpose. When I first read this post, I had an idea that should work as a more general way to approximate such a tiling for larger numbers of tiles. I don't know if this would be sufficient or not, but it seems like it might be ok for many purposes. Start with an icosahedron (the regular polyhedron with 20 triangular sides). value of n. E.g.: /\ /\ /\ ---- ---- ---- /\ /\ /\ /\ -------- -------- ... /\ /\ /\ ------------ 1 4 9 ... (The illustrations aren't so great, but remember that some of the triangles are upsidedown.) The resulting construction is scaleable, allowing meshes of size 20n^2 for all n>=1. One nice thing is that the decompositions of triangles are very regular, allowing efficient storage (exercise: find an elegant way of packing the n^2 triangles into an nXn array) as well as ease in computing the neighbor relation within large triangles. Since there is a small finite number (20) of the latter, neighbors across boundaries can be computed efficiently using table lookup. The only disadvantage I can think of is that the structure is not uniform, and weird effects may occur at the vertices of the icosahedron, where 5 triangles meet instead of 6. It still sounds like it might be fun to play with. Does anyone have any objections or comments that I haven't considered? -- Paul Callahan callahan@cs.jhu.edu