SQUARE TILINGS WITH PRESCRIBED COMBINATORICS

Let T be a triangulation of a quadrilateral Q, and let V be the set of vertices of T. Then there is an essentially unique tiling Z = (Z(v): v epsilon V) of a rectangle R by squares such that for every edge [u,v ] of T the corresponding two squares Z(u),Z(v) are in contact and such that the vertices corresponding to squares at corners of R are at the corners of Q. It is also shown that the sizes of the squares are obta ined as a solution of an extremal problem which is a discrete version of the concept of extremal length from conformal function theory. In t his discrete version of extremal length, the metrics assign lengths to the vertices, not the edges. A practical algorithm for computing thes e tilings is presented and analyzed.