# Minimal surfaces

There is also a material called microlam, which is a high grade, thick, sort of plywood used for beams. I found a piece some years ago and carved a few bowls out of it, all was well except wood is different than stone or metal in that it can catch fire when you are not paying attention.

On the discussion of removing material that is not there, the approach is not one of disliking 3d printing. This is kind of like saying that, if I am going to build a machine, I am only going to use wrenches, and no screwdrivers; of course that would be foolish, both wrenches and screwdrivers are tools, and you use one or the other depending on what you need to do. Same for 3d printing, or subtractive machining, or material forming without addingn or subtracting. different tools and techniques for different purposes and results.

On the ‘way of thinking’, removing the material that isn;t there, many sculptors and artists will do what is called a study piece on a complicated design. You do a small piece, work out the 'tool paths" (which can be a mill cutter, a hammer and chisel, the angle of a grinding wheel, the angle of the blacksmith anvil relative to the work and your hand held hammer, the angle of the molten glass bead on the end of a blow tube, etc) and then go on to create the real piece. During the process of doing a ‘study piece’, you learn about the material you need to remove, and from there you teach yourself about the shape you are trying to create.

So, to the calculation problem at hand, a) You can try modeling the material you are trying to remove; b) you can model then actual surface you want to create, with zero thickness c) you can take surface from B and add thickness, radius the edges,

Also, you might want to grab yourself a piece of wood and a file, or clay, and have a try at forming the approximate shape by hand. You will create an ugly study piece indeed. try it again, ti will be slightly better, still ugly. try a third time, and it may still look pretty bad. So what, in the process of the hand forming you will teach yourself about the shape you are creating, and wanting to model.

My design is in the ugly-study-piece phase!

If you look at the original artists other works there are some wooden versions that look like they were building up to the final limestone masterpiece.

BACK ON TOPIC…

I started looking at how you do create a minimal surface from a trefoil knot wire and found this…
Soap Films On Knots

Anyone know how to create the initial surface that could then be Soap-Filmed in K2 to get a minimal surface?

TREFOIL_KNOT_MINIMAL_SURF.gh (7.7 KB)

The more I look at that wire trefoil knot, the more I can see it in the original sculpture…

One way of creating torus knot curves is to sweep a point through a Möbius transformation (with a rational input for Q)

This also reveals the connection between the familiar representation of the trefoil with 3-fold rotational symmetry and the 2-fold one in the image above. One comes from Q=2/3 and the other from Q=3/2
You can even use another rotation of the 3-sphere to transition between them:

Then you can create an initial mesh for a Seifert surface by choosing a projection and filling in the enclosed regions with mesh faces in a chequerboard pattern, including twists where the edges cross.

However, coming back to this particular Nat Friedman stone sculpture - I’ll repeat what I said earlier in this thread - despite the title of the piece, the sharp edges do not appear to form a knot at all.
I haven’t seen any images of it from other angles, but assuming it is symmetrical, so that with a half turn around the vertical axis it would look the same, the curve is something like this:

Which you can see is possible to deform into a simple unknotted loop without passing through itself.

If instead the curve was like this it would be a knot:

…but in the photo you can clearly see that the edges in the middle cross the other way like the unknot.

…and again, it does not appear to be a thickened solid from one minimal surface, but rather a volume contained between 2 distinct surfaces.

With only this angle it is difficult to say for certain what happens to the surface at the circular boundary we see. It could be we are seeing something positively curved (and therefore not minimal) at a tangent, but I suspect it actually has another circular crease here too, like I showed in the cut view I posted earlier in this thread. Then these curves are connected by a pair of surfaces, each with a handle. Put it together and you have something like this:

I didn’t match the exact shape of the crease curve here, but my guess is that this is the right topology for the sculpture, judging from this single view.

I like the sculpture, but I think here the artist has taken inspiration from the forms of trefoil knots, minimal and Seifert surfaces, but actually made something different.

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I’m not disputing your explanation of the initial sculpture… I agree, although I can’t manage to replicate something quite as smooth as your rotating image above…

I do want to try and create a surface / mesh from the knot (or not!) curve that I can then soap film in K2. Just out of interest.

I’ll have a go at projecting the curve and filling in the enclosed regions.

Quite nice software for visualising Seifert…
SeifertView

Yes, SeifertView is nice. Also KnotPlot.
I tried a while ago some knot relaxation by repulsion, so you can input a knot as any old messy curve, as long as it has the right topology, and find the smooth energy minimising form of it:

knotrepulsion.gh (18.2 KB)

Seifert’s algorithm is quite simple. It should be possible to make something to automatically generate them in Grasshopper.

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It looks like it creates a network of discs and ribbons with twists in the ribbons according to the topology of the knot. I’m not sure why it seems to create solid meshes… possibly just to colour the 2 sides of the mesh different colours.(?)

There is a repulsion setting (Strong or Weak): I presume this is what holds the mesh apart and stops it just coollapsing in on itself?

Trefoil…

Figure 8…

Borromean…

Awesome and cool tool, thanks Daniel.
I used this script in order to make the knotted curve

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Unfortunately, the InterconnectPoints component doesn’t work on my pc… must be a 32bit thing

You can replace it with a cross reference component, with the same set of points going into A and B, and data matching set to strict lower triangular. This gives you the connection of all points to each other without duplicates or connections to themselves

Thank you

What’s the smoothing strategy? Is it possible with the K2 Goals or does it need more custom goals?

So far, using the custom repulsion goal from Daniel above and LineLength, I’ve designed Seifert’s underpants…

I’ve tried to give the naked edges more tension but as usual I seem to be missing something.

Constant Tension on the naked mesh edges plus the repulsion C# goal on the naked points transforms Seifert’s underpants…

Still not quite right in terms of mesh structure.

You need something for the internal edges too. Simplest would be Length goals with restlength of 0.

When I do that, this happens…

Smooth instead of Line seems to help but not if I subdivide the mesh…

Starting Mesh…

If I subdivide the initial mesh more…

this happens…

trefoil_relax.gh (22.0 KB)

Here’s how I approached this. I was separating the generation of the curve from the mesh relaxation.
It is possible to do them both together, but then the forces keeping the knot curve smooth/repelled from itself need to be strong enough to resist the pulling in of the surface. I’d do this with Length goals and perhaps some bending too.

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Oh also, I included a little bonus in that last definition

Because the trefoil curve is swept out by a 4d rotation, you can apply that same 4d rotation to the surface, and it inverts while staying on the same boundary

Play with the slider to see this transformation

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Any chance of a screen shot of the definition so I can fill in the gaps when opening in R5 please?

Could you try something for me if possible… I noticed that relaxing the Trefoil from 2 different starting meshes in Seifert View gave different results…

gives a result similar to yours above…

but using the flat starting mesh…

gives this…

Which looks like the inspiration for the Nat Friedman sculpture but with the knot in the edge.

Any idea why they would relax differently?