I have a lot of links embedded prettily in 3-space, and I want to look at Seifert surfaces on them.
I can flatten them into knot diagrams and find the braid word etc and use the standard algos, but then how would I pull the result back onto my original nice embedding?
Or is there any method of building Seifert surfaces on links in 3D without relaxing and flattening them into canonical posture?
@henryseg@bathsheba@nervous_jesse Indeed, level set surfaces of the solid angle (aka generalised winding number) of curves is something I've been excited about and developing tools for working with for a few years now. I posted about it a lot back when I was active on Twitter. Conventional methods for meshing implicits fail on these functions, but I've developed something that works well. I'm actually close to releasing these tools for use in Rhino.
@demofox I wonder if the tolerances there are tight enough to capture this detail. In the true optimal packing for 10, one of those 3 that look like they're in a line is actually slightly separated from the wall
The empty space between these prisms forms a connected network. Imagine deflating this space until it collapses to a network of lines - we will take this as one of our skeletal graphs: https://skfb.ly/oTOnW
We can then make a smooth triply periodic surface dividing these two graphs, as shown in the first post. Here's a larger patch of the surface on its own: https://skfb.ly/oTNZq
Now I'm not certain an exactly minimal surface with this topology exists, though initial numerical experiments seem to suggest that it might. I've not seen a surface with this topology documented anywhere (such as Ken Brakke's collection of TPMS https://kenbrakke.com/evolver/examples/periodic/periodic.html), though it's possible I've missed something.
To make clearer the relationship with the rod packing, here's a model showing 4 cylinders with their radii increased so they slightly overlap, and the negative space between these. https://skfb.ly/oTOrB
(this might be interesting to make as a physical model, since it is all developable surfaces which could be cut from flat sheet material)
@robinhouston
I've not seen it studied for quarter circles, but I believe that as with other shapes there will be no general rule or closed form solution, just a matter of finding the optimum for each case.
Here's a quick go at the first few:
@littlescraps You're the second person to be reminded of this. I watched a lot of TNG growing up but don't remember this episode. I'll have to watch it!
Henneberg's surface is a non-orientable algebraic minimal surface with self intersections.
By cutting it with 2 linked trefoil knots and one figure-eight knot, these self intersections can be removed.
A compact circle packing is one in which if we draw lines connecting the centres of each pair of tangent circles, these lines form a triangulation (equivalently - the gaps between them all have only 3 sides).
If we allow only 2 different radii, there are exactly 9 possible pairs which allow a compact packing in the plane. This delightful paper from Fernique, Hashemi and Sizova shows that for 3 radii there are exactly 164 possibilities.
Something I noticed when experimenting with my minimal surface solver is that many well known minimal surfaces are not stable.
For instance with a helicoid on a pair of helical boundary curves, it becomes unstable when the pitch goes below a critical value, and the surface flips to a ribbon around the outside (it changes from twist to writhe).
In this paper https://arxiv.org/abs/1602.02652
they show that you can have stable regions of these 2 ways, separated by a topological defect which transitions between them,
and this can act as a soliton and move along the curves (as shown in my first animation in this thread)
(on a small side note, I've noticed that looping animations display with an unfortunate slight stutter here on Mastodon, at least in the desktop browser. This would happen on Twitter when uploading them as mp4, but not when uploading as gif (even though these were converted to mp4). Is there any way to avoid this stutter here?)
