If we look at the exclusion zone of a vertex in a euclidean unit distance graph we find that it has a clearly defined measure, and as shown in the first entry on the exclusion zone, the dimensionality of this measure is (n-1)-dimensional. For where m>1 the measure is n-dimensional.

will be a n-sphere with radius 1, and will be a n-ball with radius m.

Now let’s take a look at how color has to behave in this system. One of the first things to notice is that, from a point of intuiton, since has a clearly defined measure, using a finite amount of colors to fill these colors should have a clearly defined measure as well, but a proof remains elusive.

If we do take a color and concentrate it in a continuous segment of we get a measurable line. We can disperse a color finer but there are problems with doing that:

Same colored points can have their exclusion zone overlap in 2 points only if they are close enough. Note that any point between v1 and v2 will have its exclusion zone overlap with . As we concentrate more points denser their exclusion zones overlap more. This is also what we found with areas bounded by Jordan curves, any point inside of such a single colored boundary will have it’s exclusion zone be only a subset of the exclusion already generated by the boundary.

An efficient tessellation of space has both the exclusion zones of

Conversely if we take some infinite amount of colors and disperse them finer across their combined exclusion zone within becomes bigger.

I would like to say that I have some more concrete math of what’s happening, but I don’t and I don’t have any solid ideas for how to get there either as of now.