Koch Snowflake
Start with an equilateral triangle. On each edge, replace the middle third with an equilateral bump. Repeat. The perimeter grows without bound while the area converges — infinite coastline around a finite island.
Perimeter: 3 · (4/3)^n · s | Area converges to 8/5 of original triangle | Dimension: log4/log3 ≈ 1.2619
About this lab
The Koch snowflake, first described by Swedish mathematician Helge von Koch in 1904, is one of the earliest fractal curves. The construction is deceptively simple: start with an equilateral triangle, divide each edge into three equal segments, replace the middle segment with two sides of a smaller equilateral triangle (pointing outward), and repeat.
At each iteration n, the number of sides multiplies by 4 and each side shrinks to 1/3 of its previous length. The perimeter after n iterations is P(n) = 3 · (4/3)^n · s, where s is the initial side length. Since 4/3 > 1, the perimeter grows without bound as n increases. In the limit, the Koch snowflake has infinite perimeter.
The area, however, converges. At each step, the new triangles added contribute a total area of (3/9) · (4/9)^(n−1) times the original triangle’s area. Summing the geometric series, the total area converges to exactly 8/5 (= 1.6) times the area of the initial triangle. An infinite perimeter encloses a finite area.
This is a mathematical version of the coastline paradox, identified by Lewis Fry Richardson: the measured length of a coastline depends on the length of the measuring stick. Shorter sticks reveal more detail and produce longer measurements, without limit. Benoit Mandelbrot used this observation to develop fractal geometry, assigning non-integer dimensions to such curves. The Koch curve has a fractal (Hausdorff) dimension of log(4)/log(3) ≈ 1.2619 — more than a line (dimension 1) but less than a plane (dimension 2).