The Koch Snowflake · Fractal Fair

A curve of infinite length enclosing a finite area — one segment becomes four, forever.

History

Helge von Koch introduced his curve in 1904 to give an explicit, purely geometric example of a continuous curve that is nowhere differentiable — answering critics who felt Weierstrass’s earlier analytic example was an unmotivated formula rather than a genuine “curve.”

Construction

Divide a segment into thirds. Replace the middle third with two sides of an equilateral triangle pointing outward. Repeat on every resulting segment, forever. One segment becomes $N=4$ , each scaled by $r=1/3$ :

D = \frac{\log 4}{\log 3} \approx 1.2619

If $P_n$ is the perimeter after $n$ steps from a unit triangle, $P_n = 3\cdot(4/3)^n \to \infty$ — infinite length — yet the enclosed area converges to a finite limit.

★Cool Fact

The Koch curve is nowhere differentiable: at every point, the curve has no well-defined tangent, because at every scale it turns a corner.

PLAY → NOTICE → EXPLAIN

Infinite perimeter, finite area

For the Koch snowflake, every segment becomes four segments one-third as long. The perimeter therefore multiplies by 4/3 every round, while the added triangular areas get rapidly smaller.

Iteration 4

Perimeter, relative to start3.1605×Pₙ/P₀ = (4/3)ⁿ → ∞

Area, relative to initial triangle1.6419×Aₙ/A₀ → 8/5 = 1.6

The paradox is only apparent. Boundary length and enclosed area measure different things. The edge becomes endlessly wrinkled, so its length diverges; the new bumps occupy a convergent total area.

Python implementation

Pseudocode first

Algorithm in one glance Recursive

Replace one segment by four

koch(p, q, depth) {
    if (depth == 0) return {p,q}
    divide p--q into thirds; raise the middle third into a 60-degree peak
    return the four smaller Koch curves joined end to end
}

The executable version keeps the same shape; shared plotting/export details live in fractalfair_helpers.py.

from fractalfair_helpers import path_figure

def koch(p, q, depth):
    if depth == 0:
        return [p, q]
    step = (q-p)/3
    a, b = p+step, p+2*step
    peak = a + ROTATE_60 @ step
    return (koch(p,a,depth-1)[:-1]
            + koch(a,peak,depth-1)[:-1]
            + koch(peak,b,depth-1)[:-1]
            + koch(b,q,depth-1))

λThe Same Idea in Haskell

kochPoints :: Int -> P2 Double -> P2 Double -> [P2 Double]
kochPoints 0 p1 p2 = [p1, p2]
kochPoints depth p1 p2 =
  init (kochPoints (depth-1) p1 a)
    ++ init (kochPoints (depth-1) a peak)
    ++ init (kochPoints (depth-1) peak b)
    ++       kochPoints (depth-1) b p2

Koch snowflake — **Figure:** Depth 4 (three sides of 256 segments each), coloured along the traversal from deep blue to pale cyan.

Anecdote

Koch’s curve, the Cantor set, and Weierstrass’s function together formed what Poincaré dismissively called a “gallery of monsters.” Mandelbrot’s contribution, decades later, was arguing the monsters were better models of mountains and coastlines than any smooth curve.