Discovered by Giuseppe Peano in 1890, Peano curves are one dimensional curves that densely fill higher dimensional space.
The curves depicted are the Hilbert Curve in 3D (upper left), the Mandelbrot Quartet Curve (upper right), and the 2D Hilbert Curve (bottom).

The images of the curves at the top were created by Miriam Huntley, Rob Scharein, and I (upper left), and Jian Li (upper right).
The Origami fold at bottom was created by Jason Ku and photographed by Erik Demaine.
Click on an image to get to a high-resolution version.