MINIMAL SURFACE V2 - Or just a Knot problem?

So for fans of this thread Minimal Surfaces and this Gallery thread Minimal Surface Sculpture in Oak please take a look at this challenge brought to my attention by @alex7


I’ve seen this before somewhere (or something similar but can’t remember where so if anyone knows who to credit please do.

Both myself and @alex7 have had a go at this (myself in GH and Fred in Rhino) and we can get something similar but, as with the previous discussion, getting that finesse is difficult.

This done in Rhino from lofting section curves…

The shape appears to have an edge that is a knot and an inner ellipse and outer circle. The knotted edge is not just a standard torus knot or mobius knot - it’s stylised somehow to work with the outer circle and inner ellipse…

In this attached GH file (29.1 KB)
I’ve tried to create the knot edge and the inner ellipse and outer circle and then create sections to loft but I get the usual mobius loft issues where the section curves don’t align properly.

My intention was to then create a mesh I could relax or apply a Kangaroo soap film to for a nicely curved surface.

I think I need more control over the knot to achieve the aesthetic of the original 2d sketch.

I also think this could be approached like @DanielPiker did the original Minimal Surface shape…

…But this would still require pulling the mesh to the right curve.

I think my attempt is a bit clunky and there must be a more elegant way of creating this shape.

The bits highlighted where the knot curve intersects the inner and outer ellipses are annoying me too as these determine whether the shape is 4 spiral surfaces or one mobius band…

For me the motivation is a shape that I want to hold. There was something about the original sculpture by Nat Friedman that was perfectly balanced and this shape could have the same quality…
In the last discussion we were able to get things that were topologically right but didn’t have the “je ne c’est quoi” which has made me think a lot about how to control that torus knot curve and then ultimately manipulate the final surface through smoothing, soap film or however to be able to search for a shape I want to hold enough to find a nice piece of wood and CNC it.

It’s been 3 years since the last one… Look forward to seeing your efforts!

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Are you using “knot” in relation to the topology of the surface, or in relation to the math of NURBS?

Why do you think this surface is a minimal surface?

Sorry “Minimal Surface” is possibly misleading. I used it more because the problem is similar to a previous thread about a similar shape where we explored minimal surfaces amongst many other things. I think you could use some kind of minimal surface or mesh relaxation to achieve this shape though.

I believe the edge of the shape in the 2d image at at the start follows a torus knot so it sort of is in relation to both the topology and the maths of the NURBS curve but the curve has been created by calculating points and drawing an interpolated curve through them, not a NURBS curve from control points.

For clarity the term “knot” is used as the name of a specific set of parameter values and points or isocurves in the math of NURBS curves and surfaces. All NURBS curves and surfaces have knots independent of their topology.

FYI, our goal is to achieve the very first image (2D line graph). In my Rhino model, i deliberately opened the centre area to make a new form from the original. As you can see the model itself is doable in Rhino but we are interested to get it done with GH

There are several varieties about, including Adobe Stock: (Geometric Line Art Tutorial.Halftone circles, dots pattern, vector, grunge. Comic texture background. Monochrome half-tone. Circle halftone Dots, White and blue geometric gradient for pop art designs. Stock Vector | Adobe Stock)

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Not “knot” as in NURBS knot. “Knot” as in a knot in a piece of rope which can also be described mathematically using formulae or created, for example, using the Mobius Transformation component from @DanielPiker 's Kangaroo or a torus knot or trefoil knot etc.

Trefoil Knot

The edge of the original shape follows a trefoil knot I think!

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Nice looking shape!
I will try and spend more time on this when Council work eases off again!

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