Lévy C Curve

Visualization

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

Overview

The Lévy C curve (also called the Lévy dragon) was studied by French mathematician Paul Lévy in 1938. It is a self-similar curve with the remarkable property that multiple copies of itself tile the plane without overlap. Its name comes from its resemblance to the letter C at every level of magnification.

Definition

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

The translation $(\tfrac{1}{2},\tfrac{1}{2})$ arises as $g^{-1}\cdot[0,1] = \frac{1}{2}(0+1,\,-0+1) = (\tfrac{1}{2},\tfrac{1}{2})$ — an integer input vector producing an exact rational output with no floating-point approximation.

Properties

• Fractal dimension: $\frac{\log 2}{\log \sqrt{2}} = 2$ (the attractor has positive area)
• Tiling: Four copies of the curve tile a square
• Number of transforms: 2
• Self-similarity: Each half of the curve is a scaled rotated copy of the whole

Related

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:

References

• Lévy, P. (1938). Les courbes planes ou gauches et les surfaces composées de parties semblables au tout. J. École Polytechnique.
• Lévy C curve — Wikipedia
• Riddle, L. Classic IFS — Lévy Dragon. Agnes Scott College.
• Hinsley, S. R. Self-Similar Tiles — Quadratic Pletals (Gaussian integer Z[i] family).

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