Tuesday, March 22, 2016

Art and Mathematics - The Four Color Theorem

We all remember middle and high school history courses and the hours spent coloring maps. I remember there were restrictions about having to color each adjacent countries a different color. In 1852, a young man named Francis Guthrie noticed that when coloring the counties in England, only four colors were needed. He then wondered if this is true for any map. This started a chain of events that lead to its proof more than 120 years later!

There have been several proofs offered by notable mathematicians but all had been insufficient for a formal proof. It wasn't until 1976 when two mathematicians, Kenneth Appel and Wofgang Haken used a computer to finally prove the four color theorem by showing that there are certain number of map configurations and that all required no more than 4. Now, you may be thinking what a waste of time. But this is remarkable in that this was the first time a computer was used to prove a mathematical conjecture. 

If you have ever tried to color in a coloring book or better yet at a restaurant coloring the placemat given to your child, you're going to need at least four colors to make sure that whatever you're coloring will have distinct color for each outlined area. And next time you're wondering how to color an abstract drawing like the following, just use this simple diagram..

Let's pretend that you want to color the following:

You picked your 4 colors and start to color and all of a sudden, you realize that you need another color so that two of the same colors won't be touching each other. Luckily, you made a copy of the original line drawing so you can start again, but you want to be smart this time and plan. Hmm, how do you plan? You can put tracing paper over your line drawing and draw a circle inside each area and connect the circles to each other where they touch.

Now, you have a simplified model of your line drawing.

All you do now is to start with one circle and color it in and add colors to the circles that connect to it. You can use the same colors as long as it does not appear in your connected circles. So it may look like this.

And now, you can color your line drawing.

For my art, it was a bit more complicated than assigning colors to specific regions because I arbitrarily added restrictions (I only wanted to use pink and purple for my petals and yellow for the center). I drew a simple cherry blossom flower and randomly placed lines around the whole drawing. And I only used four colors to color the whole thing. Although it may look better with more colors, sometimes, less is more. All graphics were made using Power Point and MS Paint

There are many excellent articles that shows more of the proof. Here is one.

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