Every day, our brains process an enormous amount of data. The human mind constantly searches for patterns in that data. This is a good thing, for the most part. There are certainly patterns among the random bits of stimuli. But this tendency to look for patterns can lead us to false conclusions – that a pair of socks is lucky, or that we have a chance of beating the house by stratagem in a casino – and can lead to wasted time, effort, or worse. Pivotal experiences in life, such as success in applying for jobs, submitting books to publishers, and finding companions are effectually random.
Each of the following pieces uses computer algorithms to simulate random processes to generate at least one element of the image.
Set 1: Markov Chains
A Markov Chain is a memoryless process – the probability of the next step being a given state depends only on the current state. You can define a simple Markov process by defining a set of states and a set of probabilities for transitions between states. The four pieces above are each instances of the same Markov process. Here, the size is chosen ahead of time, but the colors are chosen using a Markov chain.
Set 2: 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.