A very early example of a virtual mathematical sporder is Conway’s Game of Life (follow this link to read the fascinating Wikipedia entry.)
The game demonstrates the emergence of complex phenomena (including self-replication) from agents following very simple rules. It was simply a grid, with pixels that followed a few rules based on the presence or absence of adjacent pixels. The “player” simply sets up the initial arrangement of pixels and lets the game work itself out.
It was of interest to scientists in several fields, especially to evolutionary biologists, as it beautifully demonstrates the principle of emergent order.
You can even play it online.