In the world of Quintessence, the Earth is flat.  I had a lot of fun with this, dreaming up an alternate astronomy, physics, marine biology, etc. to make it work.  (Some of which I’ve saved for the characters to discover in the sequels.)  In this world, everyone has believed for ages that the Earth is round (surrounded by concentric spheres for the moon, sun, planets, stars, and then finally, heaven). Copernicus did the math, however, and discovered that everyone was wrong.

Incredibly, there is a group of people that promote the idea that our Earth is truly flat.  They call themselves the Flat Earth Society, and they seem to be perfectly serious.  You can check out their wiki to read their answers to the obvious questions: orbiting satellites, photographs from space, circumnavigation, night and day, the seasons.  Interesting stuff.