THE ASPECT RATIO OF THE TWIN DRAGON - 1/φ - proof and general equal-contraction IFS moment formulas

Aug 22, 2026

We determine the geometric aspect ratio of the Twin Dragon fractal, defined using the Gaussian integer 1+i, and show it equals 1/φ, where φ is the golden ratio. The major principal axis forms an angle −arctan(φ) with the horizontal. Using the moment fixed-point equation for the self-similar invariant measure, the paper gives closed-form formulas for aspect ratio, ellipticity, and principal-axis angle for general equal-contraction IFS. It also characterizes when the squared aspect ratio lies in a quadratic field and connects reciprocal aspect ratios to metallic ratios.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Introduction

Moment-based aspect-ratio definition

Twin Dragon covariance computation

Main results

Closed-form aspect ratio and orientation

Quadratic-field classification of the squared aspect ratio

Applications

Plane-filling tiles and parameter families

Tiles over Z[i] and metallic-ratio universality

arXiv :2604 .05010v5 [math .DS] 25 Apr 2026 THE ASPECT RATIO OF THE TWIN DRAGON IS 1/φ DMITRY MEKHONTSEV Abstract. We show that the geometric aspect ratio of the Twin Dragon—defined via the Gaussian integer 1 + i—equals 1/φ, where φ = (1 +√5)/2 is the golden ratio, and that its major axis makes angle − arctan φ with the horizontal. More generally, for every equal-contraction IFS {fk(z) = a (skz + tk)} with rotations sk ∈ S 1 (or reflections) and centred translations, we give closed-form formulas for the aspect ratio, ellipticity, and principal-axis angle of the attractor. When all rotations are trivial, the aspect ratio depends only on a and a single anisotropy parameter κ ∈ [0 , 1]; for a = 1/λ with λ in an imaginary quadratic ring, its square lies in the quadratic field Q ( √d ) where d is the square-free part of N (λ2 −1) . Since 1/φ is the reciprocal of the first metallic mean, it is natural to ask which others arise: every metallic ratio µm = (m + √m2 + 4)/2 (m ≥ 1) arises as the reciprocal aspect ratio of a plane-filling tile over Z [i]; moreover, Z [i] is the unique imaginary quadratic ring where collinear digits can produce metallic-ratio aspect ratios. 1. Introduction The Twin Dragon is a classical plane-filling fractal [5] governed by the Gaussian integer 1+i: its two defining contractions share the linear part (1−i)/2, and their translations are symmetric about the origin. Despite this simple, purely arithmetic construction, the attractor has a non-trivial shape that is not immediately obvious from the definition. Visually the Twin Dragon looks somewhat elongated—neither a square nor a thin sliver—which raises the question of its precise aspect ratio. A natural quantitative measure of shape for a planar set is the geometric aspect ratio: the ratio of standard deviations along the principal axes of its area distribution, or equivalently the square root of the ratio of the eigenvalues of the second-moment (covariance) matrix M of the uniform measure on the attractor. This definition gives the expected answers for elementary shapes: an ellipse with semi-axes α ≥ β > 0 has aspect ratio β/α (eigenvalues α2 /4 and β 2 /4, so the definition directly recovers the ratio of the semi-axes); a p × q rectangle has aspect ratio p/q (eigenvalues p2 /12 and q2 /12); and a square has aspect ratio 1 . The covariance matrix is the unique second-order shape descriptor of a probability distribution: any similarity-invariant quantity derived from second moments alone is a function of the eigenvalue ratio of M. Other notions of aspect ratio — the bounding-box ratio, the Feret diameter ratio, the eccentricity of the convex hull—require knowledge of the boundary or its extreme points, which for general IFS is not available inclosed form. The covariance definition, by contrast, is determined entirely by the Date: April 28, 2026 . 2020 Mathematics Subject Classification. 28A80, 37C45, 15A18, 11R11 . Key words and phrases. Twin Dragon, iterated function system, self-similar measure, second moments, golden ratio, metallic ratio, aspect ratio, plane-filling tile. Preprint: arXiv:2604.05010 . Zenodo: doi:10.5281/zenodo.19440753 . 2 DMITRY MEKHONTSEV invariant measure and is exactly computable for self-similar IFS via the moment fixed-point equation. The approach of characterising IFS attractors through moments of their invariant measures was introduced by Vrscay and Roehrig [13] and systematically developed in [14] . Further analysis of self-similar measures via moments and Fourier transforms appears in Strichartz [12] and Lau–Ngai [10]; however, explicit aspect-ratio formulas were not derived in those works. The key observation is that the selfsimilar measure satisfies a linear fixed-point equation in its moments, which can be solved exactly. When the open set condition holds, Hutchinson’s theorem [7] guarantees that the self-similar measure coincides with the normalised Hausdorff measure Hs | A , so the covariance matr

FAQs

How is the Twin Dragon’s geometric aspect ratio defined in this paper?

It uses the covariance (second-moment) matrix of the invariant self-similar measure on the attractor, taking the square root of the eigenvalue ratio along principal axes.

What exact value is obtained for the Twin Dragon aspect ratio?

The paper proves the aspect ratio is exactly 1/φ, where φ is the golden ratio (1+√5)/2.

How does the paper generalize beyond the Twin Dragon?

It derives closed-form formulas for aspect ratio, ellipticity, and principal-axis angle for general equal-contraction iterated function systems with centered translations, and it connects reciprocal aspect ratios to metallic ratios via tile constructions over Z[i].

Aug 22, 2026

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