Scherk meets Enneper

My little excursions into the history of minimal surfaces continues with a contribution of Heinrich Scherk from 1835. Making assumptions that allowed him to separate variables in the so far intractable minimal surface equation, he was able to come up with several quite explicit solutions, two of which are still of relevance today.

Scherksimple

In its simplest version, the singly periodic Scherk surface looks from far away like two perpendicular planes whose line of intersection has been replaced by tunnels that alternate in direction.

The next milestone concerning these surfaces took place 1988, over 150 years later, when Hermann Karcher constructed astonishing variations. Among others, he showed they can be had with (many) more wings

Scherk9

and even twisted:

Scherktwist

Now, can they also be wiggled? The prototype here is the translation invariant Enneper surface. It has the feature that it can be wrapped onto itself after sliding it any distance.
In other words, it is continuously intrinsically translation invariant.

TransEnneper

Hmm. I should patent this.

So we can switch out the boring flat Scherk wings with the wiggly Enneper wings, like so, still keeping everything minimal, pushing the notion to its limits.

Enneper scherk1

Here is a more radical version. You don’t want to run into this in the wild.

Enneper scherk2

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