Space-filling curve

In mathematical analysis, a space-filling curve is a special kind of curve that can reach every point in a higher-dimensional area, like the unit square or more generally, an n-dimensional unit hypercube. This means that even though a curve normally seems like a thin, one-dimensional line, a space-filling curve can cover an entire two-dimensional space!

The idea of space-filling curves began with Giuseppe Peano, an Italian mathematician. He was the first to discover such a curve, and because of this, space-filling curves in the 2-dimensional plane are sometimes called Peano curves. The term "Peano curve" can also refer specifically to the example of a space-filling curve that Peano himself found.

There is also a related idea called FASS curves. FASS stands for "approximately space-Filling, self-Avoiding, Simple, and Self-similar" curves. These curves are finite approximations of certain space-filling curves, meaning they try to get close to filling a space but in a more manageable way.

Definition

A curve is the path a point follows as it moves smoothly. In 1887, a mathematician named Jordan described a curve as a continuous function with input values between 0 and 1.

Usually, the points of a curve are in a flat plane or in 3D space. Sometimes people talk about all the points the curve reaches. Curves can also be defined without clear start or end points.

Main article: Continuous function
Main articles: Domain (mathematics) and Unit interval
Further information: Topological space and Euclidean space

History

In 1890, Giuseppe Peano found a special kind of curve that goes through every point in a square. This curve is called the Peano curve. Peano wanted to show that a line can fill up an entire area, which was a surprising idea.

After Peano, others used his idea. David Hilbert made his own version of the curve and was the first to draw a picture to help people understand it better. These curves are interesting because they show how something that looks one-dimensional can cover a two-dimensional space.

Outline of the construction of a space-filling curve

Space-filling curves are special paths that can reach every point in a shape, like a square. To build one, we start with the Cantor space, a special set of points, and create a function that maps this space to a line segment. We can then use this function to map pairs of points from the Cantor space to points inside a square.

The final step is to extend this mapping so it covers the whole square, creating a path that visits every point inside. This idea shows how paths can fill areas completely!

Main article: Cantor space

Cantor function | Cantor set | homeomorphic | compact | Tietze extension theorem

Properties

A space-filling curve can reach every point in a square or other shape, even though it looks like a single line. This is because the curve touches or overlaps itself without leaving gaps.

These curves are special types of fractals, which are shapes that repeat themselves at different sizes. They can twist a lot, and their length can grow very big while staying in a small area.

The Hahn–Mazurkiewicz theorem

The Hahn–Mazurkiewicz theorem helps us understand which spaces can be reached by a continuous curve. It says that a space can be covered by a smooth line if it is compact, connected, locally connected, and second-countable.

These special spaces are sometimes called Peano spaces. Some versions of the theorem may use the term "metrizable" instead of "second-countable," but these descriptions mean the same thing here.

Kleinian groups

In the study of Kleinian groups, there are special curves that can fill a sphere. Research by Cannon & Thurston in 2007 showed that a certain circle in the infinite part of a geometric space can act like a sphere-filling curve. This circle belongs to the infinite sphere of hyperbolic 3-space, showing how complex shapes can cover other areas completely.

Integration

Mathematician Wiener showed that space-filling curves can help simplify complex calculations. He explained that these special curves can turn hard problems into easier ones. This helps us study higher-dimensional spaces using simpler, one-dimensional methods.

Wiener pointed out in The Fourier Integral and Certain of its Applications that space-filling curves could be used to reduce Lebesgue integration in higher dimensions to Lebesgue integration in one dimension.