Is it possible to create a solid whose sides are all quadrilaterals, and whose vertices are all bringing together four edges? I've been playing around in autocad and haven't been able to come up with a way to do this. I'm trying to make a checkerboard type of solid where no touching faces have the same color.
Yes, a torus. See http://en.wikipedia.org/wiki/Euler_characteristic for a way to explore this type of question.