The Chromatic Number of the Plane
What is the minimum number of colors needed to color the plane so that
no two points of the same color are exact one unit apart? If you
look at the figure above, you can see that the answer is at most seven
colors. A few unit circles have been included on the diagram.
If you look at the widget below, you'll see that answer is four colors
or more. Just toss it onto the plane. It's not hard to prove
that at least four colors are necessary.
I made a discovery relating to this topic -- a
new coloring of the plane in seven colors.
A widget that shows that the chromatic number of the plane is at least
Thus, we know that the chromatic number of the plane is either 4, 5, 6,
or 7 at this point. It's still an open question. See Graph
Theory Open Problems, Mathematical
Coloring Book, and Geometry
Junkyard references for a little more on the history of this problem.
Alex Soifer says his Mathematical Coloring Book will be finished by the
end of 1999, which isn't far off. One object mathematicians desire
is a unit graph that requires five colors. So far, no-one has found
one. For fun, perhaps you'd enjoy finding a four-coloring for the
following unit graph.
Is this graph four-colorable?