Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!samsung!munnari.oz.au!bruce!dbrmelb!davidp
From: davidp@dbrmelb.dbrhi.oz (David Paterson)
Newsgroups: comp.graphics
Subject: Re: Tesselating the sphere
Message-ID: <765@dbrmelb.dbrhi.oz>
Date: 5 Mar 90 00:37:28 GMT
References: <155@tacitus.tfic.bc.ca> <18000002@yoyodyne>
Organization: CSIRO, Div. Building Constr. and Eng'ing, Melb., Australia
Lines: 35

> I have been playing around with various 3-D objects and have come upon,
> probably not for the first time, the fact that a dodecahedron might be the
> best polyhedron to perform simulations on.  Dodecahedrons seem to lend 
> themselves to this because they have six-sided faces which can be broken down
> into six equilateral triangles,

Ugh! Regular dodecahedrons have five-sided faces. Rhombic dodecahedrons
have four-sided faces.

> which themselves can be broken down with the 
> quad-trees approach.  The benefit of this is that all the intersections have
> six connections.  This would seem to eliminate the artifacts generated on
> tetrahedron and octahedron surfaces.

You can never surface a sphere with triangles in such a way that all
intersections have six connections. The best you can do (when you have
more than 30 triangles) is to surface a sphere in such a way that all
intersections have either n or m connections. Max(n,m) is greater
than or equal to 6. Min(n,m) is less than 6. So you can have n=5,m=6
or n=5,m=7 or n=3,m=6 or n=3,m=12 etc.

For instance. Start with a tetrahedron. Then cut each triangle up into
4,6,9,16,25 etc. triangles gives n=3,m=6. Or start with an icosahedron
and cut each triangle up into 4,6,9,16,25 etc. triangles gives n=5,m=6.
Or start with a dodecahedron, cut each face into 5 triangles and cut
each triangle up into 4,6,9,16,25 etc. triangles also gives n=5,m=6.
Solutions with n=5,m=6 have triangles that are most nearly equilateral
but are harder to program than, say, cutting up the faces of an
octahedron to get n=4,m=6.

-----------------------------------------------------------------------------
David Paterson,
CSIRO Division of Building, Construction and Engineering,
Highett, Victoria, 3190, AUSTRALIA
davidp@dbrmelb.dbrhi