## Random Walks and the Four Color Theorem

I’ve been fascinated with random walks since I heard of them as an undergraduate in mathematics. In essence, a random walk is a series of steps where after each step, a new direction is chosen at random. A famous example is the thought experiment of a Drunkard’s Walk — imagine a character just drunk enough not to know the way home, but still able to wander around, who decides to get home by choosing what way to go at each intersection by flipping a coin. What is the probability she’ll end up back at the bar?

On average, how far away from the bar can she go? The lines separating color regions in these painting are a modified Drunkard’s Walk performed on a hexagonal grid.

Speaking of the colors – a well-known property of maps (well known, at least among cartographers and mathematicians) is that you can if you want to color them in so that bordering regions are different colors, the maximum number of colors you’ll need is four (try it!). This result is easy to test on a case-by-case basis, but to prove that it really works is actually quite difficult. The mathematical formulation of the question is called the Four-Color Theorem, and was originally posed in 1852. Thirty years later, two mathematicians separately published proofs of the theorem that lasted a decade before getting proven wrong. A controversial proof-by-computer appeared in the 1970s, and finally a formal proof that appears to hold up to scrutiny appeared in the mid 1990s – 140 years after the simple problem was stated!

I’m in the middle of a series of paintings based on these principles. The first two are *Maps
*

and *Collision*,

both of which are for sale at Fine Art America.com.

Link to Maps on FAA

Link to Collision on FAA

These and more like it at geometric art for sale (Link to gallery on Fine Art America).