Why a snowflake has an infinite edge
2026-05-13
Start with an equilateral triangle. Look at any of its three sides. Find the middle third of that side and pop a smaller triangular bump out of it. Do the same to every side. The triangle now has twelve sides, each one a third as long as the originals. Now do it again - find the middle third of every one of the twelve sides, pop a bump out, repeat.
After six steps the snowflake has 12,288 sides. The numbers get out of hand quickly. The shape, oddly, does not.
Each tick of the depth counter triggers a new round of bumps. The first round turns 3 sides into 12. The second turns 12 into 48. The third into 192. By depth 6 the snowflake's edge has been chopped into twelve thousand tiny zigzags. And yet the figure still fits neatly inside the same circle the original triangle did. The faint dashed triangle behind the snowflake is the starting shape - watch it stay put as the snowflake grows around it.
Why the edge can't stop growing
One step of Koch replaces every edge with four pieces. Each piece is a third as long as the edge it came from. The total new edge length is therefore 4 × (1/3) = 4/3 of the old length. Every iteration multiplies the total perimeter by 4/3.
After N iterations, perimeter = 3 × (4/3)N. The factor (4/3) is greater than 1, so as N grows the perimeter grows without bound. There is no upper limit. By iteration 100 the perimeter is already longer than the diameter of the observable universe.
Why the area can't keep up
Each bump is itself a small equilateral triangle. At iteration 1, three bumps are added, each (1/3) the side length of the original - so each has (1/3)2 = 1/9 the area. Three of them adds 3/9 = 1/3 of the original area.
At iteration 2, every one of the twelve new sides grows its own bump. That's 12 bumps, each (1/9)2 = 1/81 the original area. Total: 12/81 = 4/27. At iteration k, the bumps number 3 × 4k-1, each with area (1/9)k of the original triangle. That's a geometric series with ratio 4/9 - less than 1 - so it converges.
The total area, divided by the original triangle area, sums to exactly 8/5. The snowflake's area is finite. It is 1.6 times the area of the triangle you started with, and that's all it will ever be.
A coastline argument
The Koch snowflake was invented by Helge von Koch in 1904 as a pathological example - a curve "continuous everywhere, differentiable nowhere" - to settle a dispute about whether such things could exist. They could, easily. He drew one.
Sixty years later Benoît Mandelbrot pointed out that real coastlines do this too. Measure Britain's coast with a 100-km ruler and you get some number. Measure it with a 1-km ruler and you get a much larger number, because you trace bays and headlands the longer ruler skipped. Measure with a 1-metre ruler and you get larger still. The finer your ruler, the longer the coast - because the coast has bumps inside bumps inside bumps. A coast is roughly a Koch curve. Its length is not a fixed number; it depends on how closely you look.
The snowflake is the simplest version of this paradox you can hold in one hand. A neat geometric shape, easy to draw at any depth, with an edge that nobody could ever finish measuring.
Three bumps. Then twelve. Then forty-eight. The perimeter multiplies by 4/3 forever, the area converges to 8/5, the figure stays inside the same circle. Finite area, infinite edge - the simplest fractal that won't stop growing.