Heighway Dragon

Fractal dimension: 2

Boundary dimension: ≈ 1.5236

Visualization

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AIFS program

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

Overview

The Heighway dragon curve was first investigated by NASA physicist John Heighway and later popularised by Martin Gardner and Donald Knuth. It is most famously associated with the paper folding sequence: repeatedly fold a strip of paper in half in the same direction, then unfold each crease to a right angle. After infinitely many folds, the result is the dragon curve.

Definition

The IFS lives in $\mathbb{Z}[i]$ (the Gaussian integers), controlled by $x^2+1=0$ . The companion matrix $s$ (defined as $companion([1,0]) in AIFS) satisfies $s^2+1=0$ , so $s \leftrightarrow i$ in the projected plane. The expansion $g = s+1 \leftrightarrow 1+i$ has $|g|=\sqrt{2}$ , so $g^{-1}$ contracts by $\frac{1}{\sqrt{2}}$ .

Map	AIFS	Complex form	Rotation
$f_1$	`s*g^-1`	$\frac{i}{1+i} = \frac{1+i}{2}$	$+45°$
$f_2$	`[1,0]s^2g^-1`	$\frac{-1}{1+i} = \frac{-1+i}{2}$	$-135°$

Note: $s^2 = -1$ , so $s^2 \cdot g^{-1} = \frac{-1}{1+i} = \frac{-1+i}{2}$ .

The only difference from the Twin Dragon is the second map: s^2*g^-1 (rotation $-135°$ ) here versus s^3*g^-1 (rotation $+135°$ ) there.

Both maps share the fixed common image $f_1(1,0) = f_2(1,0) = (\tfrac{1}{2}, \tfrac{1}{2})$ .

Properties

Fractal dimension: 2 (the attractor fills a region of the plane)
Tiling: Four copies of the dragon tile the plane
Number of transforms: 2, both contraction ratio $\frac{1}{\sqrt{2}}$
Paper folding: The $n$ -th iteration corresponds to $n$ folds of a paper strip

Paper Folding Connection

The sequence of fold directions when unfolding (R = right, L = left) follows the recurrence:

$a_{2^k(2m+1)} = \begin{cases} R & \text{if } k \text{ is even} \\ L & \text{if } k \text{ is odd} \end{cases}$

This regular paper folding sequence encodes the entire structure of the dragon curve.

Boundary Dimension

The boundary $\partial A$ has Hausdorff dimension

\dim_H(\partial A) = \frac{2\log\lambda}{\log 2} \approx 1.5236,

where $\lambda$ is the unique positive real root of $\lambda^3 - \lambda^2 - 2 = 0$ ,, $\lambda \approx 1.6956$ .

The neighbour graph of the dragon has a unique boundary component whose spectral radius satisfies the same cubic — giving the Moran equation $\lambda^2 = 2^d$ and hence $d = 2\log\lambda/\log 2$ . This dimension coincides with that of the Twin Dragon boundary, since both IFS share the same neighbour graph structure over $\mathbb{Z}[i]$ .

All four of the following IFS share the same algebraic skeleton — two maps contracting by $\frac{1}{\sqrt{2}}$ over $\mathbb{Z}[i]$ with expansion $g = 1+i$ — differing only in the power of $s$ for the second map and the choice of translations:

IFS	Second map rotation	Translation
Lévy C Curve	$-45°$ (`g^-1`)	$(0,0)$ and $(\tfrac{1}{2},\tfrac{1}{2})$
Heighway Dragon	$+135°$ (`s^2*g^-1`)	$(0,0)$ and $(1,0)$
Twin Dragon	$-135°$ (`s^3*g^-1`)	$(0,0)$ and $(1,0)$