A piece from the gallery
The Koch snowflake
Start with a triangle. Put a little pointed bump in the middle of every edge. Then do the same to every new edge, again and again. What do you think you'll end up with?
One line becomes four
Here's the whole rule. Take any straight edge and split it into three equal parts. Lift the middle part into a small pointed bump, like a tent, with two sides the same length as the parts beside them. One line has become four.
Now do that to every edge. Then to every edge of the result. That's it. There's nothing else to it.
Meet a fractal
Zoom in on any small stretch of the snowflake's edge and you'll find a tiny copy of the bigger pattern. Zoom in again and the same shape is waiting there too.
Fractal: a shape whose small parts look like the whole. The Koch snowflake, first described by the mathematician Helge von Koch in 1904, is one of the oldest and simplest examples.
You've probably seen fractal-like shapes outside math too: the branching of a fern, the outline of a coastline, frost creeping across a window.
An edge that never ends, on a shape that fits on the page
Every step swaps each straight piece for four pieces, each a third as long. So the edge gets 4/3 as long every time. After 4 steps it's about 3 times as long as it started. After 10 steps, nearly 18 times. Keep going and it grows without limit.
And yet the snowflake never spills out of its corner of the page. Its bumps only ever get smaller, so the whole shape stays snugly inside a circle around the first triangle. Its area settles down too, to exactly 8/5 of the starting triangle.
The surprise: an edge that goes on forever can wrap around an area you could color in with one crayon.
Grow your own
Change the rule and watch what happens. There's no wrong answer, only new shapes.
192 straight pieces, together 3.16 times as long as the starting edge.
Things to try
- Turn the steps up one at a time and watch the number of pieces multiply by four.
- Flip the bumps inward. You get an "anti-snowflake" that folds in on itself.
- Start from a single line. That's the Koch curve, the snowflake's original form.
- Slide the bump angle up toward 80° and watch the edge crinkle until it nearly fills the space.
Where next?
If you liked watching one small rule pile up into something intricate, try giving instructions to a turtle that only knows how to go forward and turn.