3-COLOR DIAGRAM CHASE Painting

This work illustrates a proof technique from theoretical computer science, known as Newman’s Lemma. One studies systems in which it is known that if we go one step out in two different directions, it is possible to make the ends meet again in a number of steps. The question is whether this is still possible, even if we start by going out multiple steps in two different directions. The answer is that this is not always possible. The pattern resembles constructions by the Dutch artist Escher, e.g. made popular in Hofstaedter’s book Gödel, Escher, Bach. The different squares and rectangles are painted with many different colors for aesthetic reasons. But the title refers to the fact that three colors suffice, if we want to avoid painting adjacent shapes with the same color. In the painting it is not too difficult to identify a pattern which can be repeated indefinitely with just three colors. The pattern proceeds from left to right, from top to bottom, in the diagonal, etc. The sufficiency of three colors is a special case of a more general construction, which says that one can paint an arbitrary real or fabricated map of countries with just four colors and still be able to avoid giving adjacent countries the same color. Whether this was true, was a mathematical problem that was studied for many years until the advent of computers gave new possibilities. An approach was identified which reduced the infinitely many different possible maps to a large finite number of cases, each of which was checked by computer. The proof prompted some debate whether the approach with a computer really lived up to the traditional characteristics of a mathematical proof. This work is part of the series THEOREMS AND PROOFS. The works illustrate – directly or indirectly – a theorem with a beautiful proof, according to the criteria put forth by the mathematician Hardy in his classical book A Mathematician's Apology: surprise combined with something inevitable, economy and depth.