The Four Color Theorem holds true (this was proved in the last 50 years or so), so given any map, it is possible to color the regions in such a way that no two regions that share a border have the same color. Actually finding such a coloring, on the other hand, can be quite difficult.

There is a small problem with real maps—namely, we often insist that disjoint regions be constrained to share a color (i.e. we might insist that both the lower and upper peninsulas of Michigan have the same color). The Four Color Theorem does not hold in this circumstance. That said, the US map is sufficiently free of such cases, and can be colored with only four colors (in fact, this is true of most real maps). I think that I managed to fix my error, and that the new map works.

Alternatively, I suppose that we could do a better job of choosing the characteristic functions in the first place. [latex]cos(\pi z/2) = \pm 1[/latex] when [latex]z[/latex] is even, and 0 otherwise; and [latex]\sin(\pi z/2) = \pm 1[/latex] when [latex]z[/latex] is odd, and 0 otherwise. Squaring these gives us the behaviour we want for the characteristic functions, and application of the power reducing formulae will give us the formula I originally gave.

The morals of the story: first, the smooth extensions of functions to larger domains need not be unique; and second, I shouldn’t try to work from three sets of notes at the same time.

Thanks for keeping me honest,
xander

cos(pi*z) behaves the same on positive integers as -1^z, so I guess the choice is arbitrary. I was just a little confused by the derivation. Using -1^z I get a different fractal that diverges almost everywhere.

Great post! Fun thinking about this.

Thank you,
xander