Space-filling curves

Definition

Space-filling curves are continuous, surjective functions that map a one-dimensional interval onto a higher-dimensional space, such as a plane. They demonstrate how a single continuous curve can completely cover a 2D area or higher-dimensional space.

5 Must Know Facts For Your Next Test

1. The first known space-filling curve was the Peano curve, introduced by Giuseppe Peano in 1890.
2. Space-filling curves are often constructed using recursive methods and fractal geometry.
3. These curves have applications in optimizing data storage, image processing, and solving differential equations.
4. Hilbert's curve is another popular example of a space-filling curve and is used in various practical applications.
5. Despite being continuous, space-filling curves are not differentiable due to their highly intricate structure.

Review Questions

• What is the primary characteristic of a space-filling curve?
• Can you name two types of space-filling curves and describe their main properties?
• How do space-filling curves relate to parametric equations in calculus?

Related terms

Fractal: A complex geometric shape made up of patterns that repeat at every scale and can be described recursively.

Parametric Equation: An equation where the coordinates are expressed as functions of one or more parameters.

Peano Curve: The first discovered space-filling curve that maps the unit interval $[0,1]$ continuously onto the unit square $[0,1] \times [0,1]$.