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Tori Tori Tori
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Bob underwood
Jan 19, 1999, 3:00:00 AM1/19/99
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No, thats not what i meant. Ill put it another way: If a torus can be
colored with seven colors and also formed with a minimum of seven
planar polygons (faces) , has it been proved that the number of colors
is also the same as the minimum number of flat polygon
faces required to form the model? This question applies to tori
in particular and to other manifolds in general,
colored with seven colors and also formed with a minimum of seven
planar polygons (faces) , has it been proved that the number of colors
is also the same as the minimum number of flat polygon
faces required to form the model? This question applies to tori
in particular and to other manifolds in general,
John Conway
Jan 20, 1999, 3:00:00 AM1/20/99
to
No it hasn't. What IS true, is that if the coloring number is N,
then there's always a map with just N faces that can only be colored
by making them all different.
But essentially nothing is known about which maps can be made with
planar faces, whether they're of this type or not. A little bit more
is known about the dual problem, namely that for
genus 0 1 2 3 4
the minimal number of vertices for an embedded polyhedron is
4 7 8 10 11,
and the big problem here is whether the next entry is 12. These are
lower bounds for the minimal face numbers, but as far as I know, they
haven't yet been shown to be achieved for genus 2 or more.
John Conway
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