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Newsgroups: comp.infosystems.gis
Date:         Fri, 28 Aug 1992 09:34:03 EDT
              <GIS-L@UBVM.BITNET>
From:         David_Douglas <DHDAD@ACADVM1.UOTTAWA.CA>
Subject:      Tellelation of the sphere:

There is no known way to evenly tesselate a sphere after the
ichosahedron. This was discovered by Plato, verified by Fuller and
Dutton. (Good company eh! Geoff.) There are ways to distribute a non
stratified random distribution of points on the sphere. On the plane
it is possible to use an iterative generation procedure
to take a random distribution and incrementally shift all points
until, in the end, they organize themselves into either a square grid, a
triangular lattice or a hexagonal close packing. I wonder if this same
procedure could be applied to randomly distributed points on the sphere.
If it could it would generate a now completely unknown pattern.
One would begin by generating a fine random distribution
points on the sphere. If this procedure works,  and if it is done with a
number of points other than 4 6 8 12 and 20, and if the resulting
pattern is a regular distribution or lattice on the sphere, the points
equidistance apart and the dual of the point distribution,  ie the cell
structure) has    cells all the same area and the same shape, then
whoever did it would have a seat in the mathematical annals along with
Plato, Pythagoras, Euclid, and whoever it was who statistically proved
the four colour map theory with a computer.  It will be a most
magnitudinous advance.

Anyway, that first step. A note some years ago from me to Hrvoje
Lukatela ...

What if we generate a random distribution of short
(maybe fixed length) bars
in a three d cartesian system, then immerse our sphere in that. Then find
the intersections of the bars with the surface
of the sphere. Would that not
be a random distribution of points on the sphere?

Hrvoje Lukatela is probably  the clearest thinker we have on these
matters.
He offers this as   his answer.

A random set of location on the surface of a sphere can be created as
follows:

1) Obtain three random real numbers between -1.0 and 1.0 .  Treat them
as cartezian coordinates of a point inside a cube with a side of 2.

2) Reject the point if the magnitude of its radius-vector is greater
than 1.

3) Reject the point if the magnitude of its radius-vector is smaller
than 1 - epsilon.

4) Normalize the raduis-vector of the points passing above tests, and
treat them as vector coordinates on the surface on the sphere.

Tests should, of course, be carried out in the square measure.

Selection of epsilon:  Small epsilon (i.e.  accepting only the points
very close to the sphere's surface) will protect well against the source
random number generator's imperfection in the vicinity of 0 - which is
scaled-up in the normalization; this at the expense of the algorithm's
yield.  0.5 seems like a good starting point.

This is indeed the immersion of the sphere in a uniformally-populated
cartesian space...

Will stay in touch...

Regards, Hrvoje

Incidentally the hierarchical tesselation of the triangles of one of the
platonic solids will not produce a regular tellelation. We can see this
visibly in any Buckminster fuller geodesic dome. You can see the
patterns adjust themselves to fit every so often. These adjustments are
a few of the more visible uneven triangles.
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