As someone who is interested in math, physics, and computer science, I often come across very intriguing topics that, at first glance, appear completely irrelevant to one another. It was just last year when I learned about the idea of emergence and was finally able to see an underlying correlation between the topics.
For example, things like the game Go, fractal geometry, and even genetic algorithms can all be studied from an emergent point of view. Each of these is an example of exceedingly complex patterns/systems which are created from a simple set of rules over a period of time/epochs. If we consider a typical game of Go, the arrangement of the stones can seem chaotic and random. However, the end game-state is achieved by playing the game and observing the simple set of rules. With fractals, complex patterns at the large scale are formed by repeated use of geometric patterns at the smaller scale. In computer science, genetic algorithms were born from the study of cellular automaton (i.e. Conway's Game of Life). Genetic algorithms are used for search/optimization problems. By mimicking biological evolution, algorithms are mutated, combined, and evolved over a period of time until the most 'fit' algorithm is attained. Such techniques have been used by companies to create algorithms which predict stock prices. Pretty neat.