Jason (jcreed) wrote,

I remembered this morning reading something about how the four-color theorem is equivalent to a certain peculiar "associativity" theorem about quaternions: that for any two parenthesizations of n variables, there is some assignment of unit quaternions i, j, k that in both parenthesizations makes no intermediate result 1.

Here is a visualization for n=8 of how many of these mutual assignments (which amount to edge 3-colorings of pairs of binary trees joined at the leaves, which is a little bit closer to cell 4-colorings of planar graphs):

Darker pixels mean more possible assignments. The 4-color theorem is the statement that throughout the triangle, it never gets completely white. Just think! If you could come up with a simple closed form description of this sort of diagram, you might have a nice short proof of 4CT.
Tags: math

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