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Thu Jan 11 16:47:43 CET 2007

LED lights on a circle


I ran into an interesting geometrical or topological problem.  Working
with N-simplexes to drive LED lights using a minimal amount of
electrical nodes.  The problem I have is how to associate an N-symplex
with a circle in a natural way.  For a tetrahedron this is trivial,
but what with higher dimensional things?

It's an interesting problem, because it seems to make absolutely no
sense to do so.  There should be no reason why a higher dimensional
polytope should be naturally projectable on a 2-sphere.  But i can't
refute it just like that.

To rephrase the problem in a more down-to-earth wording: I want to
create a 2-sphere covered with lights in a way that the lights are
normally distributed, and i have a certain pattern to access them.  At
the same time, i'd like to do this with as little connections as
possible.

Note that simplexes are easier to work with as they follow the
binomial expansion for vertices, edges, triangle faces, tetrahedral
volumes, etc...

Trying the simple thing of projecting a 5-plex on a sphere. Compared
to the 4-plex (tetrahedron), there is one pair of crossing lines.



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