Koch Snowflake

Visualization

@
$dim=2
$subspace=[s,0]
s=$companion([1,-1])
r=$exchange()
g=s+1
h0=s*r
h1=s*[0,1]
h2=s*[1,0]
h3=s*[-1,0]
h4=s*[0,-1]
h5=s*[-1,1]
h6=s*[1,-1]
A=(g^-1*h0|g^-2*h1|g^-2*h2|g^-2*h3|g^-2*h4|g^-2*h5|g^-2*h6)*A

Overview

The Koch Snowflake is one of the earliest described fractals, introduced by Helge von Koch in 1904. It is a filled plane region bounded by the Koch curve, a continuous nowhere-differentiable curve with Hausdorff dimension $\log(4)/\log(3) \approx 1.2619$.

The snowflake itself — the bounded closed region — has Hausdorff dimension $2$, filling a solid area with a fractal boundary.

Seven-Map Decomposition

The snowflake decomposes into seven self-similar pieces:

• $h_0$: one central copy at scale $1/\sqrt{3}$, which is a rotated (reflected) version of the whole
• $h_1$–$h_6$: six outer copies at scale $1/3$, arranged with 6-fold rotational symmetry

This satisfies the dimension equation:

$\left(\frac{1}{\sqrt{3}}\right)^2 + 6 \cdot \left(\frac{1}{3}\right)^2 = \frac{1}{3} + \frac{6}{9} = 1$

confirming Hausdorff dimension $d = 2$.

Algebraic Structure

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) is the companion for $x^2 - x + 1$, whose roots are $e^{\pm i\pi/3}$ — rotation by 60°.

The expansion matrix $g = s + 1$ corresponds to multiplication by $1 + e^{i\pi/3} = \sqrt{3}\,e^{i\pi/6}$ (scaling by $\sqrt{3}$, rotation by 30°), so $g^{-1}$ contracts by $1/\sqrt{3}$ and $g^{-2}$ by $1/3$.

The reflection matrix $r$ ($exchange() in AIFS, [[0,1],[1,0]] in $\mathbb{R}^2$) represents complex conjugation in the projected plane.

The Koch Curve Boundary

The boundary of the snowflake consists of three copies of the Koch curve, each built by the classic four-map IFS:

$f_1(x) = \frac{x}{3}, \quad f_2(x) = R_{60°}\frac{x}{3} + \frac{1}{3}, \quad f_3(x) = R_{-60°}\frac{x}{3} + \frac{1}{2}, \quad f_4(x) = \frac{x}{3} + \frac{2}{3}$

with Hausdorff dimension $d = \log(4)/\log(3) \approx 1.2619$.

References

• von Koch, H. (1904). Sur une courbe continue sans tangente. Arkiv för Matematik, Astronomi och Fysik.
• Koch snowflake — Wikipedia
• Riddle, L. Classic IFS — Koch Snowflake. Agnes Scott College.
• Hinsley, S. R. Self-Similar Tiles — Koch Snowflake tiling (trihextal derivative).

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