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February 21st, 2017, 02:52 PM   #1
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Cool Novel fractal inspired loop - Analytical or level set representation?

Hi all, I hope this is of interest to some of you and there is someone out there that is able to either provide an answer or direct me along the path to finding it myself. I will keep this to a length I hope is sufficient and necessary.

I am an applied mathematician working part time on a novel geometry producing an interesting elastic effect. I have created 3D printed prototypes which I can physically test. Numerically I hope to use a Finite Element Analysis tool that is currently designed to use 'level set geometries' as opposed to Computer Aided Design STL files.

This idea of level set is somewhat new to me but my understanding is that it is a function that takes at point in space and produces a scalar output, <0 if it is outside of the geometry, =0 on the surface and >0 if within.

I need analytical representations of geometry to progress in this avenue. An example analytical representation was given to me for the torus (or unknot):

I will show pictures of the geometry in the hope that someone can provide a function to describe it.

This is the first 3D print. Its a closed loop equivalent to the unknot (a series of twists could return it to the unknot). Each (out of plane) crossing has an over-under pattern.

The curvature of each part is defined by the a radius mapping between two vertices of a triangle. I hope the construction lines in this drawing makes it clear:

The size of each curve/triangle in the branching get smaller by 1/1.618 - this ratio could be anything <1 but the choice of the golden ratio is relevant in the next part.

If this is taken as 'iteration 3' of a branching knot, further iterations give convergence to a fractal pattern which is a Koch curve variant with non-uniform scaling - or the Golden Symmetric Binary Tree.

I am primarily interested in an analytical representation of iteration 3 forming a 'smooth' closed surface and a level set representation. I dont know if this could be easily extended to iteration 4 or iteration n? My best guess is that this could be achieved through some smooth local plane warping from a base torus model.

I am aware that it may be possible to produce this as three separate parts which are tessellated to produce the final shape. I mainly want to know if this can be done in one go, but am open to any ideas.

Disclaimer: work shown here is original and is hoped to be developed into an open-source project. All questions/ feedback is welcome.
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February 22nd, 2017, 01:48 AM   #2
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Your 3d creation is quite beautiful. I wonder if you can fill the tube with something to make it glow (like a neon sign) ?

Sorry I have no math help to offer

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February 22nd, 2017, 04:21 AM   #3
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Do you need explicit parameterizations of sublevel and superlevel sets? It doesn't appear you do so why not just interpolate?
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February 22nd, 2017, 05:16 AM   #4
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That's cool by the way, how do you determine the radius of the outer loops at each vertex on the outside of the shape? Is it an independent parameter (like the size reduction factor for the triangles constructions at the vertex)?
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