In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, a space-filling curve is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
whose
range contains the entire 2-dimensional
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 ...
(or more generally an ''n''-dimensional unit
hypercube). Because
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 ...
(1858–1932) was the first to discover one, space-filling curves in the
2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the
Peano curve, the specific example of a space-filling curve found by Peano.
Definition
Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion,
Jordan
Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...
in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a ''curve'':
In the most general form, the range of such a function may lie in an arbitrary
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, but in the most commonly studied cases, the range will lie in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
such as the 2-dimensional plane (a ''planar curve'') or the 3-dimensional space (''space curve'').
Sometimes, the curve is identified with the
image of the function (the set of all possible values of the function), instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the
real line (or on the open unit interval ).
History
In 1890,
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 ...
discovered a continuous curve, now called the
Peano curve, that passes through every point of the unit square. His purpose was to construct a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
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 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 ...
. Peano was motivated by
Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same
cardinality as the infinite number of points in any finite-dimensional
manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e., a curve that fills a space. Peano's solution does not set up a continuous
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the unit interval and the unit square, and indeed such a correspondence does not exist (see below).
It was common to associate the vague notions of ''thinness'' and 1-dimensionality to curves; all normally encountered curves were
piecewise differentiable (that is, have piecewise continuous derivatives), and such curves cannot fill up the entire unit square. Therefore, Peano's space-filling curve was found to be highly counterintuitive.
From Peano's example, it was easy to deduce continuous curves whose ranges contained the ''n''-dimensional
hypercube (for any positive integer ''n''). It was also easy to extend Peano's example to continuous curves without endpoints, which filled the entire ''n''-dimensional Euclidean space (where ''n'' is 2, 3, or any other positive integer).
Most well-known space-filling curves are constructed iteratively as the limit of a sequence of
piecewise linear continuous curves, each one more closely approximating the space-filling limit.
Peano's ground-breaking article contained no illustrations of his construction, which is defined in terms of
ternary expansions and a
mirroring operator. But the graphical construction was perfectly clear to him—he made an ornamental tiling showing a picture of the curve in his home in Turin. Peano's article also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to
graphical visualization was motivated by a desire for a completely rigorous proof owing nothing to pictures. At that time (the beginning of the foundation of general topology), graphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counterintuitive results.
A year later,
David Hilbert published in the same journal a variation of Peano's construction. Hilbert's article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. The analytic form of the
Hilbert curve, however, is more complicated than Peano's.
Outline of the construction of a space-filling curve
Let
denote the
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
.
We start with a continuous function
from the Cantor space
onto the entire unit interval