Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!rphroy!caen!uwm.edu!bionet!agate!eos!jbm From: jbm@eos.arc.nasa.gov (Jeffrey Mulligan) Newsgroups: comp.graphics Subject: Re: recursive sphere tesselation from tetrahedral seed Message-ID: <8079@eos.arc.nasa.gov> Date: 16 Apr 91 21:50:32 GMT References: <1991Apr15.171109.2625@rice.edu> <3221@borg.cs.unc.edu> Distribution: usa Organization: NASA Ames Research Center, California Lines: 34 leech@homer.cs.unc.edu (Jonathan Leech) writes: >In article <1991Apr15.171109.2625@rice.edu>, fontenot@comet.rice.edu (Dwayne Jacques Fontenot) writes: >|> I have been playing with recursive triangular sphere tessellations. Starting >|> with an octahedron works beautifully... >|> But, the tetrahedral tessellation is giving me some strange results. >|> My question is this; is there a fundamental problem with starting with a >|> tetrahedron? I cannot think of one... any replies will be greatly appreciated. > I think it's just a bit odd-looking for the first few levels due >to the odd symmetry of a tetrahedron compared to an octahedron, which >has 3 perpendicular symmetry planes. It looks fine by the 3rd or 4th >subdivision level. All of the added vertices will have a valence (degree) of 6, i.e., 6 edges will meet at the vertex. The original vertices of the polygon will retain their degree, which for the octahedron is 4, and for the tetrahedron is 3. After a large number of recursive subdivisions, the triangular facets meeting at the original vertices of the octahedron will be right equilateral triangles, while those meeting at the vertices of the original tetrahedron will be obtuse equilateral triangles (angle =120 deg). The most uniform tesselation is obtained from the icosahedron, the vertices of which all have degree 5. In the limit, the facets at these vertices will be triangles having angles 72, 54 and 54 degrees (want 60 60 60). -- Jeff Mulligan (jbm@eos.arc.nasa.gov) NASA/Ames Research Ctr., Mail Stop 262-2, Moffett Field CA, 94035 (415) 604-3745