This Is Not a Helicoid

But almost. It has a vertical axis, lots of horizontal lines, and it twists.

Nothelicoid

But it is part of something bigger, a triply periodic minimal surface. 32 copies of the above piece, replicated by rotations and reflections, look like this:

Nothelicoidcopies

This surface sits in a rectangular box over a square. If you identify top and bottom edge of the original squarical helicoid, you get a doubly twisted annulus, which is intimately (confomally, that is) related to a hollow spiderweb:

D spider 01

 

 

When squeezing the height down, our non-helicoids become even more helicoidal. When pulling the height up, the helicoids disappear. What we have here is a deformation of the Diamond surface of Hermann Amandus Schwarz.

When he sees this, he will probably just nod.

One

What happens when we pull a little further? We see doubly periodic Scherk surfaces emerging, stacked on top of each other.

Triplyscherk

 

 

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