Koch Curve

Visualization

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

Overview

The Koch curve was introduced by Swedish mathematician Helge von Koch in 1904 as an example of a continuous curve that is nowhere differentiable. The familiar Koch snowflake is obtained by applying the construction to all three sides of an equilateral triangle.

Construction

Starting from a unit line segment, at each step replace the middle third with two sides of an equilateral triangle (pointing outward), removing the base:

$\text{segment} \;\longrightarrow\; \text{four segments of length } \tfrac{1}{3}$

After $n$ steps the curve has $4^n$ segments each of length $(\frac{1}{3})^n$, so the total length grows as $(\frac{4}{3})^n \to \infty$.

Definition

The IFS lives in $\mathbb{Q}(\omega_6)$ where $\omega_6 = e^{i\pi/3}$ is a primitive 6th root of unity. The companion matrix $s$ (defined as $companion([1,-1]) in AIFS) satisfies $s^2 - s + 1 = 0$; in the projected plane it acts as rotation by 60°.

The expansion $g = s + 1$ has $|g| = \sqrt{3}$ and $g^2 = 3s$, so $g^{-2} \cdot s = \tfrac{1}{3}I$ — a pure contraction by $\frac{1}{3}$ with no rotation.

The four maps factor as $A = g^{-2}(h_1 \cup h_2 \cup h_3 \cup h_4)(A)$ with integer-coefficient operators:

The translations $[1,-1]$, $[-1,2]$, $[2,0]$ are exact integers in $\mathbb{Z}^2$; the irrational values $\tfrac{1}{3}$, $\tfrac{2}{3}$, $\tfrac{\sqrt{3}}{6}$ emerge only after projecting via $g^{-2}$, with no floating-point approximation.

The directive $subspace=[s,0] tells the renderer to project the lattice $\mathbb{Z}^2$ onto the eigenplane of $s$, where basis vector $[1,0]$ maps to $(1,0)$ and $[0,1]$ maps to $e^{i\pi/3} = (\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})$.

Properties

• Fractal dimension: $\frac{\log 4}{\log 3} \approx 1.261$
• Perimeter: Infinite
• Number of transforms: 4, all contraction ratio $\frac{1}{3}$
• Self-similarity: Each quarter of the curve is a scaled, rotated copy of the whole

References

• Koch, H. von (1904). Sur une courbe continue sans tangente. Arkiv för Matematik.
• Koch snowflake — Wikipedia
• Riddle, L. Classic IFS — Koch Curve. Agnes Scott College.

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