The colors follow the line from its start to its end. The address of the page keeps the curve, to share it.
A line with no thickness that fills a square
In 1890 Giuseppe Peano did something that seemed impossible: he described a continuous curve, a line drawn without lifting the pen, that passes through every single point of a square. Mathematicians had just learned from Georg Cantor that a line and a square have "as many" points; Peano showed that one can be drawn onto the other. A year later David Hilbert gave a simpler picture of the same idea.
Each curve is the limit of a recipe. Draw a simple path, then replace each part with a smaller copy of the whole path, turned to fit, and again, and again. Raise the Level to watch the line crowd the square: at every level it visits every cell of a finer grid. Here the recipes are written as L-systems, the rewriting rules the biologist Aristid Lindenmayer invented in 1968 to describe how plants grow.
The curves are not only curiosities. Because points close together on the line stay close together in the square, the Hilbert curve and the Z-order are used to store maps and images in databases, to dither pictures, and to divide the globe into cells (Google's S2 library uses a Hilbert curve on the faces of a cube). The dragon curve, from a strip of paper folded in half again and again, decorates the chapter pages of Jurassic Park.