Let’s start with a balance configuration with 4 helicoids with positive spin at the corners of a square, one helicoid with negative spin at the center, and one catenoid off-center, like so:
The the right is a small regeneration to twist about 2º. We can numerically twist this surface of genus 5 further:
Then the surface decides it has had enough and reverts to its initial state. Similar balance configurations for n-gons: helicoids with positive spin at the vertices with a helicoid with negative spin at the center, and a catenoid at . Why would be expect a different behavior for Below are the regenerated surfaces at 15º and 45º.
At 90º, four screw motion copies of surface becomes translation invariant:
Taking this cover increases the genus 6 of the screw motion invariant surface to a whopping 21.
The surface can be twisted further, I will update this post with more images when they become available. This, together with these examples, indicated that a larger number of helicoids is responsible to twist beyond a certain threshold. A more precise statement will follow soon.
