Here is a strange and beautiful fractal known as a dragon curve:

A dragon curve (note: this is a twin-dragon curve or Davis-Knuth dragon)

And here is the shape generally regarded as the dullest and most everyday of all:

A square

But squares are square, so let’s go back to dragon-curves. This particular kind of dragon-curve looks a lot like a Chinese dragon. You can see the same writhing energy and scaliness:

Chinese dragon

Dragon-curve for comparison

Dragon-curves also look like some species of soft coral:

In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself

Dragon-curve rep-4

Dragon-curve rep-8

Dragon-curve rep-16

Dragon-curve rep-32

Dragon-curve self-tiling (animated)

From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)

And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1

Dragon-curve tiling #2

But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square

Then you see how the shape can be replaced by five smaller copies of itself:

Square overlaid by five smaller squares

Square replaced by five smaller squares

Then you set about replacing it with two of those smaller copies:

Replacing squares Stage #0

Replacing squares Stage #1

Then you do it again to each of the copies:

Replacing squares Stage #2

And again:

Replacing squares #3

And again:

Replacing squares #4

And keep on doing it:

Replacing squares #5

Replacing squares #6

Replacing squares #7

Replacing squares #8

Replacing squares #9

Replacing squares #10

Replacing squares #11

Replacing squares #12

Replacing squares #13

Replacing squares #14

Replacing squares #15

And in the end you’ve got a dragon-curve:

Dragon-curve built from squares

Dragon-curve built from squares (animated)