Fractals

Shapes that repeat their own structure at every scale.

A fractal is a shape showing detail at every magnification, usually built by repeating a rule. Zooming in reveals structure resembling the whole rather than resolving into smooth lines.

Iteration

Classic fractals come from a simple rule applied to its own output. The Koch curve starts with a segment, removes the middle third and replaces it with two sides of an equilateral triangle, then applies the same rule to every resulting segment. The Sierpinski triangle removes the middle triangle formed by the midpoints, then repeats inside each remaining triangle.

Fractal dimension

Fractals need not have whole-number dimension. If a shape splits into N copies each scaled by 1/s, its dimension is log N / log s. The Koch curve gives four copies at one third scale, so log 4 / log 3 ≈ 1.26 — more than a line, less than a plane. The Sierpinski triangle gives log 3 / log 2 ≈ 1.58.

Infinite perimeter, finite area

Each Koch iteration multiplies length by 4/3, so perimeter grows without limit. The figure nonetheless fits inside a finite box and encloses finite area. That combination is the classic result, and it is why coastline length depends on the ruler used to measure it.