The first examples of periodic minimal surfaces with helicoidal ends (besides the helicoid itself) are Hermann Karcher’s twisted Scherk surfaces from 1988.

Here are a few of them, rendered with Bryce back in 1999.

As you can see, these can be twisted more and more so that they appear to become *two* helicoids glued together. In this case, the two helicoids turn the same way. A few years later, Martin and I were looking at more general ways of gluing helicoids together to obtain new minimal surfaces. The model case is what we called a *parking garage structure*: You can describe them mathematically as superpositions of complex argument functions, like so:

Here the numbers z(k) designate the location of the axis viewed from above, and the ε(k) can be +1 or -1, depending on the spin of the helicoid.

Note that the graph of the multivalued function arg(z) is half of a helicoid (that stays on one side of the vertical axis).

An example with three helicoidal columns of the same spin, placed at -1, 0, and 1, looks like this:

If you alternate the spin, you get surfaces that *untwist* to higher genus helicoids, we believe.

It is also possible to place the columns off a common line, like so:

Nobody knows what minimal surfaces these untwist to.

The images above were made with Mathematics in 2001. Later I found a simple way to do this in PoVRay, which I might explain next time. Here an image from 2002:

Most people get easily lost in parking garages that have only two columns. It would be cool to have a computer game where one can walk around these more complicated structures, with the location of the columns moving in time …

### Like this:

Like Loading...