Begin with an **N** shape made by connecting four points at the corners of a square like this:

Next, make a copy of the whole figure at each corner of the **N** shape:

Finally, connect the last point of one figure with the first point of the next, where “last”, “first”, and “next” are relative to the initial order of the points:

Repeating the process generates a non-self-intersecting, plane-filling curve
called the *Lebesgue curve*:

(I must be doing this sideways, since some references call this the *z-order curve*.)

The Lebesgue curve maps (one-dimensional) integers to (two-dimensional) points in a square:

Of course, there are other ways to connect the initial four points and to arrange subsequent figures. The following figures show the evolution of quasi-Lebesgue curves generated by the other possible arrangements.

Alas, randomizing the order of arrangements at each step doesn’t seem that interesting, either:

You might enjoy the 3D version of the same thing.

See, for example, Hans Sagan, Space-Filling Curves, Universitext, 1994, pp. 76–80.

Figures created with *Mathematica* 6.

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