Besides computation per se, the other big role computers play is as Muse or Sandbox, a realm to play and explore, and thereby inspire and challenge the imagination. Thus we have SimCity, we have AgentSheets and we have purpose-built simulations and models of all manner of intricate phenomenon.

If you think you understand something—ocean waves, for example, and you think the phenomenon is governed by three simple rules—well, you can't create an ocean governed by those rules and tinker with them. But you can rig a computer to follow those rules and show how they interact and unfold on screen.

There's one book I'd like to recommend for the general reader that gives a sense of how insightful this can be: The Planiverse, by A. K. Dewdney.

And if you haven't done so before, I also recommend checking out any one of the free implementations of the "Game of Life" (aka "Conway's Life") which is an elegant example of an abstract sort of simulation that sheds light on how the world might be.