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"Perfect" Klein bottle [message #91822] Thu, 12 June 2014 16:13 UTC
Hi all,

It's been a while since I posted here myself, but after five years in 3d printing it was high time I'd tackle the Klein bottle.
I been having a love/hate relation with this shape ever since I first saw it:
So elegant, yet.. something was wrong.

Well not any more!

Klein bottle - Voronoi weave:

The Klein bottle.

A mathematical classic, it was first described by the German mathematician Felix Klein in 1882 and is a great example of a non-orientable surface. It has no defined inside or outside, as the surface loops back on it self.

This is my take on one of the more popular exercises in 3d printing, hoping to complement the idea and perfection of the original surface and previous attempts before me.

Crossing borders

Because there is one thing with the physical versions I have seen so far. They lack the spirit and esthetics of the original concept and have an issue at the junction.
Whenever the bottle loops back in on itself, at the crossing, one usually sees a crude fuse/weld, or simply a blunt hole in the surface through which it passes itself.
Now for the transparent glass-blown versions that can actually function as a bottle, I think this is an acceptable compromise and I applaud the skills of the craftsmen.
But with 3d printing I felt we could do better.

Weave!
So to get around this issue, one can apply a pattern that allows for the surface to go through itself without touching or visually disturbing the flow of the surface itself.
I chose a cellular Voronoi pattern that is "self-aware":
It does not touch the passing surface, but weaves perfectly in between itself.

This model was created using a custom build tool set which utilizes Voxels.
First, I ran a charged particle simulation, which distributes the particles evenly across the endless surface of the Klein bottle, while taking into account the different "planes" of existence. Then in a voxel-space the cellular pattern is created using those particles as voronoi cell centers, but again taking into account the different planes.
And finally the model was converted to polygon format for uploading and 3dprinting.

[Updated on: Thu, 12 June 2014 16:56 UTC]

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Re: "Perfect" Klein bottle [message #92065 is a reply to message #91822 ] Mon, 16 June 2014 21:08 UTC
Awesome!
I didn't understand in your algorithm how did you manage to make voronoi cells not touching each others at the intersection ?
Re: "Perfect" Klein bottle [message #92067 is a reply to message #91822 ] Mon, 16 June 2014 21:53 UTC
Congratulations Stijn!
You've made perfect virtual self intersection of that surface.

I guess we now need to add such cool cellular surface patterns to ShapeJS as well.

Vladimir Bulatov, 3D graphics researcher, Shapeways
Re: "Perfect" Klein bottle [message #92101 is a reply to message #92067 ] Tue, 17 June 2014 10:54 UTC
two thumbs up!
Re: "Perfect" Klein bottle [message #92123 is a reply to message #91822 ] Tue, 17 June 2014 16:19 UTC
I love this elegant design variation of the Klein Bottle Opener. Next step 3D print in metal?

Social Media Specialist | Shapeways

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