I have been mulling over a possible simulation of the development of road systems, and for some reason I can't force myself to actually start writing code. I'm hoping that some math-funster will kick me out of my rut. My hope is that the model will generate realistic-looking road networks from scratch. I'm curious to see if it will capture some of the features we are used to in road networks: a preference for right-angled intersections, a fractal-ish vascularization of the area with major and minor arteries giving way to capillaries, and so on. And if it doesn't, I'm even more curious to know what kinds of networks it will generate instead. We begin with a plane region, possible a square or a disk. At the beginning of the simulation, travel speed is a fixed constant, say 1 meter per second. We select two points at random from the region, with a uniform distribution, and imagine that somebody needs to travel from one point to the other. The traveler selects the fastest possible route, and goes. The act of traveling over a route increases the speed limit along that route by one centimeter per second. After that first journey, the travel speed is 1 meter per second everywhere, except along the line between those first two points; along that line, the speed is 1.01 meters per second. We now select a new two points at random, find the shortest path by the modified metric, and perform the journey; again, the speed limit is increased by 0.01 meters per second along the chosen route. If the new endpoints lie near the original line, there will be an advantage in sidling toward that line, and scooting along it until the destination nears. (My algebra indicates that there is a critical angle at which it is most advantageous to join an existing route.) After repeating this process thousands of times, what sort of road network will have developed?