Twin Dragon

Visualization

@
$dim=2
g=[1,-1,1,1]
A=g^-1*([-1,0]|[1,0])*A

Overview

The Twin Dragon (also called the twindragon or Davis–Knuth dragon) is a self-similar fractal with Hausdorff dimension 2 — it has positive area and is a connected solid region. It is formed by joining two copies of the Heighway Dragon along their boundaries; the two copies fit together perfectly because the Heighway Dragon’s boundary is a matching fractal curve.

The Twin Dragon is a rep-2 tile: two congruent copies tile a scaled copy of itself, and by repeating this, four copies of the Twin Dragon tile the plane.

Algebraic Structure

The IFS is controlled by the polynomial $x^2 - 2x + 2$, whose roots are $1 \pm i$. The expansion matrix is:

The matrix $g$ represents complex multiplication by $1 + i$: it expands distances by $|1+i| = \sqrt{2}$ and rotates by 45°. Its inverse $g^{-1} = \frac{1}{2}\begin{pmatrix}1 & 1 \\ -1 & 1\end{pmatrix}$ corresponds to multiplication by $\frac{1-i}{2}$.

IFS Definition

The Twin Dragon $A$ is the unique non-empty compact set satisfying:

where the two maps use symmetric shifts $\pm 1$ along the real axis:

Both pre-maps are pure translations. Expanding:

Both maps share the same linear part $g^{-1}$ and differ only in translation. In complex notation:

The attractor is symmetric about the origin (maps are related by $z \mapsto -z$). Both maps contract by $\frac{1}{\sqrt{2}}$, giving self-similar dimension $\frac{\log 2}{\log \sqrt{2}} = 2$.

Properties

• Hausdorff dimension: exactly $2$ — the set has positive Lebesgue measure
• Contraction ratios: both $f_1$ and $f_2$ contract by $1/\sqrt{2}$; Moran equation: $2 \cdot (1/\sqrt{2})^2 = 1$
• Tiling: two Twin Dragons (one original, one rotated 180°) tile a $\sqrt{2}$-scaled copy; four copies tile the plane
• Relation to Heighway Dragon: the Twin Dragon is the union of a Heighway Dragon and its image under $z \mapsto -z + (2, 0)$ (the 180° rotation about the midpoint $(1, 0)$)
• Number system: the Dragon is the set of all $\sum_{k=1}^{\infty} \varepsilon_k \left(\frac{1-i}{2}\right)^k$ with $\varepsilon_k \in \{-1,\,+1\}$ — balanced binary expansions in base $\frac{1-i}{2}$, centered at the origin
• Polynomial: $x^2 - 2x + 2$ with root $1 + i$; $\det(g) = 2$

Aspect Ratio from Second Moments

The geometric aspect ratio of the Twin Dragon — the ratio of standard deviations along the principal axes of its area distribution — equals exactly $1/\varphi$, where $\varphi = \tfrac{1+\sqrt{5}}{2}$ is the golden ratio. This is surprising: the Twin Dragon is defined purely via the Gaussian integer $1+i$, with no pentagonal or Fibonacci structure in its construction.

The result follows by solving the covariance fixed-point equation for the self-similar measure. Since the similarity dimension equals 2, that measure coincides with Lebesgue area, so $M$ captures the actual shape of the attractor. The matrix $M = \int xx^T\,d\mu$ satisfies $M = A_0 M A_0^T + b_1 b_1^T$, with unique solution:

The eigenvalues are $I_{1,2} = \frac{1 \mp \frac{1}{\sqrt{5}}}{2}$, giving:

The same constant $\varphi$ also governs the orientation: the major axis of the Twin Dragon makes an angle $-\arctan\varphi \approx -58.3°$ with the horizontal, in the direction $(1, -\varphi)$.

For a proof see Mekhontsev (2026).

References

• Davis, C. & Knuth, D. E. (1970). Number representations and dragon curves. Journal of Recreational Mathematics, 3(2–4).
• Twin dragon curve — Wikipedia
• Hinsley, S. R. Self-Similar Tiles — Quadratic Pletals (Gaussian integer Z[i] family).
• Mekhontsev, D. (2026). The aspect ratio of the Twin Dragon is 1/φ. Zenodo.

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