Golden-Ratio-Based Rectangular Tilings

A golden-ratio-based rectangular tiling of the first quadrant of the Euclidean plane is constructed by drawing vertical and horizontal grid lines which are located at all even powers of phi along one axis, and at all odd powers of phi on the other axis. The vertices of the rectangles formed by these lines can be connected by rays starting at the origin having slopes that are odd powers of phi. A refinement of this tiling results in the familiar one with horizontal and vertical grid lines at every power of phi along each axis. Geometric proofs of the convergence of several known power series in phi are provided.