Square tilings with prescribed combinatorics

LetT be a triangulation of a quadrilateralQ, and letV be the set of vertices ofT. Then there is an essentially unique tilingZ=(Z_v: v ∈ V) of a rectangleR by squares such that for every edge <u,v> ofT the corresponding two squaresZ _u, Z_vare in contact and such that the vertices corresponding to squares at corners ofR are at the corners ofQ. It is also shown that the sizes of the squares are obtained as a solution of an extremal problem which is a discrete version of the concept of extremal length from conformal function theory. In this discrete version of extremal length, the metrics assign lengths to the vertices, not the edges. A practical algorithm for computing these tilings is presented and analyzed. The author thankfully acknowledges support of NSF grant DMS-9112150.