As you may know, C. S. Peirce worked for the USGS. One of the things he did while there was design a map projection that has come to be known as Peirce's quincuncial. In it one hemisphere is presented as a square, and the other hemisphere as four isosceles triangles around the sides, creating a map that is genuinely square and tiles nicely.
* Wikipedia has a good account of the basics
* Peirce's paper on it is available at Google Books. (Notice that he uses the tessellation property in the picture here.)
* Sean Carroll discussed it at Cosmic Variance a few years ago, as did John Baez.
* The same principles of projection can be used to make interesting pictures.