Universal tilings of the plane by 0-1 -matrices (Q920099)

A tiling of the plane by black and white unit squares of the square lattice is called (a,b)-universal if constructed using horizontal and vertical translations of a fundamental (m,n)-rectangle and if every one of the \(2^{ab}\) possible distinct (a,b)-rectangles of black and white unit squares occurs somewhere in the tiling. If the fundamental (m,n)- rectangle is as small as possible (implying \(mn=2^{ab})\) the tiling is called optimal (a,b)-universal. The authors prove that for all a,b\(\geq 2\) there exists optimal (a,b)-universal tilings of the plane determined by fundamental \((2^{b(a-1)},2^ b)\)-matrices with O-1 entries. Examples of optimal (3,2)- and (3,3)-universal tilings are shown in black and white, using numerously repeated (16,4) resp. (64,8) fundamental rectangles. The patch of optimal (3,3)-universal tiling exhibited reminds one (approximately) of a section of black-white marbled paper.