# Squaring the Plane

This is the homepage of our research in tiling the plane with
squares.

## abstract

We can tile the plane with exactly one of each square of positive
integer side.

You can download the paper and view a short animation (120K), a longer one (532K), or a larger version (1.7M) of the same. The VPython program used to generate the
animations is available by request, but it's still under
development.

### open questions

- Is there a "simple" tiling with squares?
- Is there a tiling using just the odd squares?
- Answer (8/06): no. If no two members of a set add to a
third, then there does not exist a tiling of the plane using only
squares with side-length from the set.
- Conjecture: If a set grows faster,
asymptotically, than the Fibonacci numbers, then it can't tile the
plane.

- Can the half-plane or quarter-plane be tiled?
- Is there a three-colorable tiling?
- Is there a tiling with triangles?
- Can the efficiency of the tiling be improved?
- Can space be cubed?