Some curves are so convoluted they wiggle free of the one-dimensional world and fill up space
How to Fill Up Space
It’s easy to sketch a curve that completely fills the interior of a square. The finished product looks like this:
How uninformative! It’s not enough to know that every point is covered by the passage of the curve; we want to see how the curve is constructed and what route it follows through the square.
If you were designing such a route, you might start out with the kind of path that’s good for mowing a lawn:
But there’s a problem with these zigzags and spirals. A mathematical lawn mower cuts a vanishingly narrow swath, and so you have to keep reducing the space between successive passes. Unfortunately, the limiting pattern when the spacing goes to zero is not a filled area; it is a path that forever retraces the same line along one edge of the square or around its perimeter, leaving the interior blank.
The first successful recipe for a space-filling curve was formulated in 1890 by Giuseppe Peano, an Italian mathematician also noted for his axioms of arithmetic. Peano did not provide a diagram or even an explicit description of what his curve might look like; he merely defined a pair of mathematical functions that give x and y coordinates inside a square for each position t along a line segment.
Soon David Hilbert, a leading light of German mathematics in that era, devised a simplified version of Peano’s curve and discussed its geometry. The illustration at right is a redrawing of a diagram from Hilbert’s 1891 paper, showing the first three stages in the construction of the curve.