Golden-Ratio-Based Rectangular Tilings

TL;DR

This paper introduces a novel golden-ratio-based rectangular tiling of the plane, explores its geometric properties, and provides proofs of convergence for certain power series involving the golden ratio.

Contribution

It constructs a new tiling pattern based on the golden ratio and offers geometric proofs for the convergence of related power series.

Findings

The tiling pattern is based on lines at powers of the golden ratio.

Vertices connect to rays with slopes at odd powers of the golden ratio.

Proofs demonstrate convergence of specific power series in .

Abstract

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.