Crochet and the two-color theorem

The entire math community is familiar with the famous 4 Color Theorem. Its statement states that any map, drawn on the plane, can be painted in 4 colors without neighboring regions having the same colors.

But there is a more humble theorem, not as robust as the previous one, but still very interesting called the “two-color theorem”. Cool, but what does this have to do with crochet?

Let’s imagine a crochet used to make flat surfaces like rugs, and with well-defined color regions. As the figure below:

We can imagine the color pattern of this rug following a structure similar to this:

We can easily, with two colors, color all its regions, without neighboring regions having the same color, as shown below with the colors red and blue:

It is also easy to find rugs that cannot be colored with just 2 colors, without neighboring regions having the same color, as in the example below:

If we generalize this mat, we will have something like this:

To see that it’s impossible to color with just two colors without neighboring regions having the same color, look at the center square. If we paint it blue. The two regions adjacent to it should be painted red. But these regions are adjacent to each other, so they would be neighboring regions of the same color.

You might be thinking, anyway, there are maps/carpets that can be painted with 2 colors and others that cannot, and nobody knows if it will be possible until they try… Wrong! This is where our not-so-famous, but important, two-color theorem comes in! It ensures that any map/carpet drawn only by interlocking closed curves can be painted with only two colors, without neighboring regions having the same colors. Below I have a little video showing some cases:

The inspiration for this post came at the exact moment I saw the following crochet mats:

At the time, I realized that they could be generalized as interlocking closed curves:

So, trying beforehand, I already knew that it was possible to color these regions with at most two colors! Now I will make this coloring with blue and red to show that it is indeed possible!

I hope you enjoyed this text 🙂

Cover image taken from instagram.com/croche.para.aprender._/