HFIHC@CUNYVM (10/05/90)
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? -Hoss ------- H. Y. Firooznia HFIHC@CUNYVM.CUNY.EDU
tim@maths.uwa.oz.au (Tim Boykett) (10/05/90)
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? >-Hoss >------- >H. Y. Firooznia >HFIHC@CUNYVM.CUNY.EDU I have never implemented one, but a number of cell-shapes and sizes could be used. My first guess was the pattern used on a soccer ball, but I seem to remember that has a mixture of hexagons and pentagons on it. You will need to use shapes that are polyhedra, ie they are topologically isomorphic to a sphere, but they have flat surfaces. There is a technical name for this process, but I can't remember it. You will need a shape for the faces of your polyhedra. The patterns to use would have to be triangles, squares or pentagons, presuming yoy want regular polygons for the faces. This is because 3 hexagons make a flat tiling. with triangles, you could have 3,4 or 5 around each vertex. 3 would give tetrahedral shape, 4 would give a pair of square-based pyramids joined at the base, and 5 would give a 20-faced object. You can see what these look like in any role-playing game shop, the are dice with 4,8,20 sides resp. With square faces, you can only get a cube shape, with pentagons you can only get a 12-sided shape. I don't think there are anyother regular polyhedra that can be made though have a weak memory of someone shoing me a 100-sided dice that was regular. I hope this semi-mathematical diatribe answers your question, otherwise don't hesitate to ask again. Tim. tim@madvax.maths.uwa.oz.au
jhanks@hubcap.clemson.edu (Jim Hanks) (10/05/90)
tim@maths.uwa.oz.au (Tim Boykett) writes: >HFIHC@CUNYVM writes: >>Has anyone ever designed and/or implemented a CA for a spherical surface? >You will need to use shapes that are polyhedra, ie they are >topologically isomorphic to a sphere, but they have flat surfaces. >what these look like in any role-playing game shop, the are dice >with 4,8,20 sides resp. With square faces, you can only get a cube >shape, with pentagons you can only get a 12-sided shape. > I don't think there are anyother regular polyhedra that can be made >though have a weak memory of someone shoing me a 100-sided dice >that was regular. You are correct in saying that that are no other regular polyhedra other than the tetrahedron, cube, octahedron, dodecahedron, and icosahedron (4,6,8,12,and 20 faces respectively). However there are other solids which are not regular but which have identical faces. For example, I have seen a 30-sided figure whose sides were rhombuses (diamonds). This solid is not regular but as each face has four neighbors, may be useful as a base for a "spherical" CA. In fact, I remember a book which had a whole slew of solids like this, but I can't remember the name :(. Something like "Mathematical Models". Hope this helps jhanks@hubcap.clemson.edu ---------------------------------- No sig yet - first post ever, actually!
callahan@cs.jhu.edu (Paul Callahan) (10/06/90)
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
bbadger@x102c.harris-atd.com (Badger BA 64810) (10/06/90)
In article <10791@hubcap.clemson.edu> jhanks@hubcap.clemson.edu (Jim Hanks) writes: >tim@maths.uwa.oz.au (Tim Boykett) writes: >>HFIHC@CUNYVM writes: >>>Has anyone ever designed and/or implemented a CA for a spherical surface? [...] The book ``Game, Set, Math'' has a chapter on mathematical models of viruses. It features the pseudo-icosahedrons -- a family of nearly spherical shapes based on replacing each of the twenty triangular faces with an {a,b} collection of triangles. For example: {1,1} 3 / \ / \ 1-----2 {2,2} 3 / \ / \ *-----* / \ / \ / \ / \ 1-----*-----2 {2,1}: 3 / \ / \ *-----*-----* \ / \ / \ \ / \ / \ *-----*-----2 / \ / / \ / 1-----* Each side of the pseudo triangle fits itself, and the ragged, but symmetrical shape can be substituted into the pattern for an icosahedraon. When you bend the flat pattern into the icosahedral shape, you must bend some triangles, if you're not using a {k,k} form. To get closer to a sphere you can project each point from the sphere's center to its surface, but the triangles be come slightly distorted. In fact, you can do the same thing with tetrahedra and octahedra, too, but the projection distortions are more severe. A cube could also be expanded into pseudo-cubes. If look at vertices instead of faces, you can get eqi-distant points from a dodecahedron by filling in each pentagon with the pattern formed from the vertex net of a dodecahedron flattened out. (But this medium isn't convenient to draw it in. :-) ---- What does a ROM say? ``Read my chips ... No new data!'' Bernard A. Badger Jr. 407/984-6385 |"Get a LIFE!" --- J.H. Conway (Just joking! :-) bbadger@x102c.ess.harris.com |Buddy, can you paradigm? bbadger%x102c@trantor.harris-atd.com |'s/./&&/g' Tom sed expansively.
schraudo@beowulf.ucsd.edu (Nici Schraudolph) (10/06/90)
Another famous family of mixed polyhedra are the Fuller spheres, perhaps
better known in various cut-off shapes as fuller domes. A soccer ball is
an example of the simplest Fuller sphere, made of 12 regular pentagons and,
uh, 20 regular hexagons. Larger shapes can be created by adding more
hexagons into the mix, at the expense of roundness: larger Fuller spheres
look more and more like dodecahedra (with the pentagons as vertices) than
spheres.
Any volunteers for working out an adressing scheme for these beasts? :-)
--
Nicol N. Schraudolph, C-014 "Big Science, hallelujah.
University of California, San Diego Big Science, yodellayheehoo."
La Jolla, CA 92093-0114 - Laurie Anderson.
nici%cs@ucsd.{edu,bitnet,uucp}lance@motcsd.csd.mot.com (lance.norskog) (10/09/90)
Send a message to: netlib@ornl.gov with this text send index from polyhedra help The polyhedra archive has coordinate systems for 143 different polyhedra. There is a man page describing the format of the databases, and referring to a library which is not in the archive. Just send for a few, have fun!