geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the Peano curve is the first example of a

space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...

to be discovered, by

Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...

in 1890. Peano's curve is a surjective, continuous function from the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...

onto In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

the

unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...

, however it is not injective. Peano was motivated by an earlier result of

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

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''th step constructs a set ''S_i'' of squares, and a sequence ''P_i'' of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, ''S''₀ consists of the single unit square, and ''P''₀ is the one-element sequence consisting of its center point. In step ''i'', each square ''s'' of ''S''_{''i'' − 1} is partitioned into nine smaller equal squares, and its center point ''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'' is chosen in such a way that the distance between the first point of the ordering and its predecessor in ''P_i'' also equals the side length of the small squares. If ''c'' was the first point in its ordering, then the first of these four orderings is chosen for the nine centers that replace ''c''.. The Peano curve itself is the limit of the curves through the sequences of square centers, as ''i'' goes to infinity.

Variants

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

Hilbert curve The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe ...

is a simpler variant of the same idea, based on subdividing squares into four equal smaller squares instead of into nine equal smaller squares.

References

{{Fractals Theory of continuous functions Fractal curves