As any mathematician who has revealed his (or her) occupation to a neighbour on a plane flight has discovered, most people associate mathematics with something akin to the more agonizing forms of medieval torture. It seems indeed unlikely that mathematics would be done at all, were it not that a few people discover the play that lies at its heart. Most published mathematics appears long after the play is done, cloaked in lengthy technicalities which obscure the original fun. The book in hand is unfortunately scarcely an exception. Never mind; after a fairly detailed introduction to the art of creating tilings and fractal limit sets out of two very carefully chosen Möbius maps, we are finally set to embark on some serious mathematical play. The greatest rewards will be reaped by those who invest the time to set up their own programs and join us charting mathematical territory which is still only partially explored.

All the limit sets we have constructed thus far began from a special arrangement of four circles, the Schottky circles, grouped into two pairs. For each pair, we found a Möbius map which moved the inside of one circle to the outside of the other.