Fold paper in half forever and you get a dragon

2026-06-09

Take a long strip of paper. Fold it in half, always bringing the right end over onto the left. Fold the folded strip in half the same way. Again. A few folds in, it is too thick to bend - but imagine you could keep going. Now unfold it completely and set every crease to a clean right angle. Look at the strip edge-on. You are looking at a fractal called the dragon curve.

Step the number of folds up and watch it grow. It looks chaotic, but it is the most orderly thing imaginable - it is literally a record of which way each crease was folded, and that record has a ruthlessly simple recursive structure.

Why folding is recursion

Here is the key. When you fold the strip one more time, you are taking everything you have folded so far and laying a fresh copy of it on top - but flipped, because the second copy travels back the other way. Unfolded into right angles, that means:

the next dragon is the current dragon, followed by a single turn, followed by the current dragon again - rotated a quarter turn and traversed backwards.

The two colours in the figure are exactly those two halves. The first half is the previous fold's dragon. The second half is the same curve, pivoted 90° about the point where they join. Every level contains two copies of the level below it. That is recursion - and unusually, it is recursion you can perform with your hands, by folding.

The curve that tiles the plane

Each fold doubles the number of segments: 2, 4, 8, 16, and on. The curve twists back on itself constantly but - remarkably - never crosses itself. And copies of the dragon fit together with no gaps and no overlaps: four dragons meet snugly around a point, and the plane can be tiled with them forever. A shape born from folding a single strip turns out to paper the whole floor.

It belongs to the same family as the Hilbert curve and the Koch snowflake: a trivial rule, applied to its own output, producing something no one would guess from the rule alone.

Every fold is a copy of every fold before it, turned a right angle and laid back down. Do it enough times and the creases remember a dragon.