Many plane-filling curves can be created by beginning with a seed shape—in this case, an S-shaped curve—and rotating or reflecting copies of the seed shape in a 3×3 grid in such a way that the end point of one copy is adjacent to the start point of the next copy. Connecting the figures and repeating the process indefinitely results in a plane-filling curve.

In the 3×3 grid, each copy of the seed shape can be oriented in one of two ways and still have its start and end points in the correct corners for the overall arrangement. The curves can thus be indexed with or 512 different values, though some curves with different indexes are identical after rotation or reflection of the whole figure.

Snapshot 1: every curve begins (iteration 0) with the S-shaped starting shape

Snapshot 2: any curve indexed by a power of 2 flips only a single copy of the shape; in this case, curve 16 flips only the center copy in the 3×3 grid

Snapshot 3: a greater iteration level for an index that is a power of 2 shows the effect of the flip on the smaller copies as well as the overall figure