Peano curve

In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.

Construction

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

i

{\displaystyle i}

th step constructs a set

S

i

{\displaystyle S_{i}}

of squares, and a sequence

P

i

{\displaystyle P_{i}}

of the centers of the squares, from the set and sequence constructed in the previous step. As a base case,

S

0

{\displaystyle S_{0}}

consists of the single unit square, and

P

0

{\displaystyle P_{0}}

is the one-element sequence consisting of its center point.

In step

i

{\displaystyle i}

, each square

s

{\displaystyle s}

of

S

i

−

1

{\displaystyle S_{i-1}}

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

c

{\displaystyle c}

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. There are four such orderings possible:

Left three centers bottom to top, middle three centers top to bottom, and right three centers bottom to top

Right three centers bottom to top, middle three centers top to bottom, and left three centers bottom to top

Left three centers top to bottom, middle three centers bottom to top, and right three centers top to bottom

Right three centers top to bottom, middle three centers bottom to top, and left three centers top to bottom

Among these four orderings, the one for

s

{\displaystyle s}

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

P

i

{\displaystyle P_{i}}

also equals the side length of the small squares. If

c

{\displaystyle c}

was the first point in its ordering, then the first of these four orderings is chosen for the nine centers that replace

c

{\displaystyle c}

.