[betterunix2]

"Does the plane have to be a plane in the mathematical sense, i.e. infinite? Because then you're obviously not going to cover it with anything. It goes on forever."

You certainly can tile an infinite plane. In fact, you can view this algebraically if it helps, by talking about tiling the complex plane. For example, the Gaussian Integers, which are complex numbers of the form (a+bi) where a and b are integers, can be viewed as the corners of squares that tile the complex plane.