Peano curve

Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality.

[2] Peano's curve may be constructed by a sequence of steps, where the

of the centers of the squares, from the set and sequence constructed in the previous step.

is partitioned into nine smaller equal squares, and its center point

is replaced by a contiguous subsequence of the centers of these nine smaller squares.

This subsequence is formed by grouping the nine smaller squares into three columns, ordering the centers contiguously within each column, and then ordering the columns from one side of the square to the other, in such a way that the distance between each consecutive pair of points in the subsequence equals the side length of the small squares.

is chosen in such a way that the distance between the first point of the ordering and its predecessor in

also equals the side length of the small squares.

The Peano curve shown in the introduction can be constructed using a Lindenmayer system.

In the definition of the Peano curve, it is possible to perform some or all of the steps by making the centers of each row of three squares be contiguous, rather than the centers of each column of squares.

These choices lead to many different variants of the Peano curve.

[3] A "multiple radix" variant of this curve with different numbers of subdivisions in different directions can be used to fill rectangles of arbitrary shapes.