Tame Twin Dragon

Visualization

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

Overview

The tame twin dragon is a self-similar tile of the plane constructed from two affine maps, both contracting by $1/\sqrt{2}$. Its attractor has Hausdorff dimension 2 — it tiles the plane with positive area.

The name distinguishes it from the classical Twin Dragon and its variants: the “tame” qualifier indicates the tile has well-behaved (simply connected, no holes) topology.

Algebraic Structure

The construction is based on the companion matrix $g$ of the polynomial $x^2 - x + 2$:

The eigenvalue of $g$ in the upper half-plane is:

This number lives in the ring $\mathbb{Z}\!\left[\tfrac{1+\sqrt{-7}}{2}\right]$ — the ring of integers of $\mathbb{Q}(\sqrt{-7})$. The norm of $\lambda$ is 2, making it a valid base for a radix number system.

The two IFS maps in AIFS are $h_0 = g^{-1} \cdot [0,0]$ and $h_1 = g^{-1} \cdot [1,0]$, with digit set $\{0, 1\}$. In the eigenvalue basis (rendered via $subspace=[g,0]) they become:

The attractor $A$ is the unique compact set satisfying:

The Hausdorff dimension equals exactly 2:

Properties

• The tile $A$ tiles the plane by translations in the lattice $\mathbb{Z}\!\left[\tfrac{1+\sqrt{-7}}{2}\right]$.
• Every point $z \in A$ has a unique base-$\lambda$ expansion $z = \sum_{k=1}^{\infty} d_k \lambda^{-k}$ with digits $d_k \in \{0,1\}$.
• It is a connected, simply-connected region with a fractal boundary.

Boundary Dimension

The boundary $\partial A$ has Hausdorff dimension

where $\lambda$ is the unique positive real root of $\lambda^3 - \lambda - 2 = 0$,, $\lambda \approx 1.5214$.

The full characteristic polynomial of the boundary substitution is $x^5-x^4+x^3-x^2-4$, which factors over $\mathbb{Q}$ as $(x^3-x-2)(x^2-x+2)$. The second factor contributes only complex roots, so the minimal polynomial governing $\dim_H(\partial A)$ is $\lambda^3-\lambda-2=0$.

References

• Bandt, C. (1991). Self-similar sets 5. Integer matrices and fractal tilings of ℝⁿ. Proceedings of the American Mathematical Society, 112(2), 549–562.
• Mekhontsev, D. (2026). The aspect ratio of the Twin Dragon is 1/φ. Preprint.

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