Dance, Phosphor Hahnemühle German Etching Print

A minimal surface is a mathematical abstraction of a soap film.

The catenoid is the surface of rotation swept by revolving the hyperbolic cosine graph \(x = \cosh z\) about the \(z\)-axis. The catenoid is minimal. Inversely, it has been known since the 18th century that (up to scaling) the catenoid is the only "complete" non-planar minimal surface of rotation.

The helicoid is the surface swept by a line \(L\) orthogonal to the \(z\)-axis in three-space if we rotate \(L\) at unit angular speed about the \(z\)-axis while simultaneously translating \(L\) at unit speed along the \(z\)-axis. Mathematicians call the helicoid a "ruled" surface because it is a union of lines. The helicoid is minimal. Inversely, it has been known since the 18th century that (up to scaling) the helicoid is the only "complete" non-planar minimal ruled surface.

Remarkably, one full turn of the helicoid can be wrapped around the catenoid without stretching or folding; there is a "local isometry" from the helicoid onto the catenoid.

More is true: The catenoid and helicoid are the real and imaginary parts of an abstract circular family of surfaces in complex three-space. When this family is projected to real three-space by rotating through the complex circle parameter and taking the real part, we get the family of minimal surfaces shown.

Printed on archival quality Hahnemühle German etching paper (310gsm) with a velvety surface.