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From: bbadger@x102c.harris-atd.com (Badger BA 64810)
Newsgroups: comp.theory.cell-automata
Subject: Re: Spherical CA
Message-ID: <4453@trantor.harris-atd.com>
Date: 5 Oct 90 19:57:33 GMT
References: <90277.175712HFIHC@CUNYVM.BITNET> <1990Oct5.064458.26401@uniwa.uwa.oz> <10791@hubcap.clemson.edu>
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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.