{"id":3301,"date":"2021-07-31T05:53:00","date_gmt":"2021-07-31T08:53:00","guid":{"rendered":"https:\/\/www.blogs.unicamp.br\/zero\/?p=3301"},"modified":"2022-05-23T17:52:16","modified_gmt":"2022-05-23T20:52:16","slug":"crochet-and-the-two-color-theorem","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/zero\/3301\/","title":{"rendered":"Crochet and the two-color theorem"},"content":{"rendered":"\n

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.<\/p>\n\n\n\n

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?<\/p>\n\n\n\n

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

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Image taken from instagram.com\/croche.para.aprender._\/<\/strong><\/a><\/figcaption><\/figure><\/div>\n\n\n\n

We can imagine the color pattern of this rug following a structure similar to this:<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

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:<\/p>\n\n\n\n

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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:<\/p>\n\n\n\n

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Image taken from instagram.com\/croche.para.aprender._\/<\/strong><\/a><\/figcaption><\/figure><\/div>\n\n\n\n

If we generalize this mat, we will have something like this:<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

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.<\/p>\n\n\n\n

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\u2026 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:<\/p>\n\n\n\n

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