I’ve had fun writing about the Hadwiger-Nelson problem to determine the chromatic number of the plane, but I think this will be my last post on the topic for now!
Of course, the original point of the hexagonal 7-coloring in my last two posts is that it establishes an upper bound of CNP (although it turns out it’s also just a really cool pattern). Again, there is a balancing act here: we have to give each hexagon a diameter of , so no two points in the same hexagon will be 1 unit apart; but we also have to make the hexagons big enough that same-colored hexagons are more than 1 unit from each other. This is indeed possible, since same-colored hexagons always have two “layers” of other hexagons in between them. Denis Moskowitz made a really nice graphic illustrating this:
In a comment on my previous post, Will Orrick pointed out that if you tile 3-dimensional space with cubes and color them with seven colors so that each cube is touching six others with all different colors, then take a diagonal slice through that space, you get this!
This is the same as the 7-colored hexagonal tiling I showed before, but with extra triangles in between the hexagons (and the colors of the triangles follow a pattern similar to the hexagons). I could stare at this all day! Here’s a version with numbers if you find it helpful. (If you support me on Patreon you can get automatic access to bigger versions of all the images I post—though to be honest if there’s a particular image you want a bigger version of, you can just ask nicely!)
What is CNP?
So, we know CNP is either 5, 6, or 7. So which is it? No one is really sure. With some unsolved problems, there is widespread agreement in the mathematical community on what the right answer “should be”, it’s just that no one has managed to prove it. That isn’t the case here at all. If you ask different mathematicians you will probably get different opinions on which number is correct. Some mathematicians even think the “right” answer might depend on which axioms we choose as a foundation of mathematics!—in particular that the answer might change depending on whether you allow the axiom of choice (a topic for another post, perhaps).