The aspect ratio of the Twin Dragon is $1/\phi$

TL;DR

This paper proves that the geometric aspect ratio of the Twin Dragon fractal equals the reciprocal of the golden ratio, revealing a surprising connection despite its purely Gaussian integer-based construction.

Contribution

It establishes a novel geometric property of the Twin Dragon related to its aspect ratio and links it to the golden ratio through covariance analysis.

Findings

The aspect ratio of the Twin Dragon is exactly 1/φ.

The covariance fixed-point equation determines the aspect ratio.

The result is surprising given the fractal's Gaussian integer origin.

Abstract

We show that the geometric aspect ratio of the Twin Dragon equals $1/\varphi$, where $\varphi = (1+\sqrt{5})/2$ is the golden ratio. The result follows by solving the covariance fixed-point equation for the self-similar measure, which coincides with Lebesgue area since the similarity dimension is 2. The appearance of $\varphi$ is surprising: the Twin Dragon is defined purely via the Gaussian integer $1+i$, with no pentagonal or Fibonacci structure in its construction.