Space-filling curve

Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube).

Related Topics:
Curve - Giuseppe Peano - Unit square

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Intuitively, a "continuous curve" in the 2-dimensional plane or in the 3-dimensional space can be thought of as the "path of a continuously moving point". To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a "continuous curve":

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A curve (with endpoints) is a continuous function whose domain is the unit interval . In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most common cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a "plane curve") or the 3-dimensional space ("space curve").

Related Topics:
Curve - Continuous function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Sometimes, the curve is identified with the range or 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 (0,1)).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Space-filling curves are curves whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). In 1890, Peano discovered a densely self-intersecting curve which passed through every point of the unit square. This was the first example of a space-filling curve.

Related Topics:
1890 - Peano

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It was common to associate the vague notions of "thinness" and "1-dimensionality" to curves; all normally encountered

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

curves were piecewise smooth (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 counter-intuitive.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~