## Minimal Surfaces, At Their Limit

The minimal surfaces in the post about Möbius strips were made using a formula by Emmanuel Gabriel Björling, a Swedish Mathematician from the 19th century. For a given curve in space, this formula allows you to write down a parametrization for a minimal surface that not only contains the given curve, but is also tangent to the curve in any way you wish to prescribe. For instance, the multiple times twisted Möbius strips all contain a circle, and touch the circle by spinning around it more or less often.

This formula works not only for circles but also for other curves, like the helix above. The difficulty is that in most cases, the equations are so complicated that they become meaningless. There are some pretty exceptions, like this knotted minimal strip:

In the search for interesting and simple curves where Björling’s formula gives manageable results, the multiply twisted minimal strips are particularly useful. We saw that the surfaces in their associate family are also closed strips, but their core curves are not just circles anymore. Using these as new core curves can be used to compute surprisingly simple formulas for surfaces like this one:

This can get quite complicated. Try to view this stereo pair cross-eyed or with a stereo viewer.

These shapes appear to contradict what we think should be a minimal surfaces. But that’s what we do: Seek what goes against the convention.