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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 $ϕ$ along one axis, and at all odd powers of $ϕ$ 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 $ϕ$. A refinement of this tiling results in the familiar one with horizontal and vertical grid lines at every power of $ϕ$ along each axis. Geometric proofs of the convergence of several known power series' in $ϕ$ are provided.