[math-fun] Shortest boundary of tiles in a tessellation of the plane

Suppose we tessellate the plane by the unit squares with vertices at the integer points. We'd now like to adjust this tessellation so that 1) The boundary of each tile is a simple closed polygon; 2) The set of translations that carry each tile to another tile is exactly the integer vectors (K,L) ∊ Z^2. Puzzle: Which tile shape satisfying 1) and 2) is such that each tile's boundary polygon has the shortest possible length? —Dan