  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.
