Caleidoscopic cell for triply periodic minimal surfaces

On some request about how the cube in some of my minimal surface projects are done i've decided to make a post for this.

So this is an example for a caleidoscopic cell, which is needed to draw a triply periodic minimal surface. In this example shows a 1/12 trirectangular tetrahedron. That is the "cage" for the so called batwing.

More information about triply periodic minimal surfaces and caleidoscopic cells you can find here...

So this should be the result:

You can divide the tetrahedron, shown above, further in order to obtain a 1/24 or even a 1/48 cell.

Here ist an example of the usage of these cells. I'll post some other projects related to triply periodic minimal surfaces later on.

Also here you can download the example cell as Rhino 5 and Rhino 4 version.

Attachments:

Caleidoscopic_Cell.3dm, 404 KB
Caleidoscopic_Cell_R4.3dm, 394 KB

Replies to This Discussion

Permalink Reply by Vangel Kukov on November 7, 2012 at 12:26am

Thanks! Appreciated!

Permalink Reply by Daniel Piker on November 7, 2012 at 5:14am

Nice project!

It probably doesn't matter for the purposes of this design, but I don't think you are actually getting a minimal surface when you join your kaleidoscopic cells.

In the same way that joining several straight line segments at their ends gives you a polyline which is not necessarily another straight line, or joining multiple arcs does not necessarily give you another arc - piecewise satisfaction of the zero mean curvature property does not guarantee global satisfaction.

Seeing the kinks in the surface at 00:19, it looks like you are fixing the boundary curve during the surface relaxation, but to get a true minimal surface, the curve itself would also need to relax. I'll post an example later of how you can fix vertices on a boundary face, but still allow them to move on that face to avoid this.

Permalink Reply by Florian Frank on November 7, 2012 at 6:36am

Thanks, this would be very interesting. Thats true...in fact in terms of calculation time, result that we were searching for and form control we decided to do a relaxation at the end.

Later on in the project there are no more kinks because of that.

The general question is: Is the relaxation method an enough precise calculation method for a real minimal surface?

Permalink Reply by Daniel Piker on November 7, 2012 at 10:57am

Relaxation using 1D springs gives an approximation of a minimal surface, but the result will be affected by the meshing. It can be very close provided the meshing is reasonably even. What counts as 'precise enough' depends on your application though.

The next release of Kangaroo will contain a true membrane element, that calculates forces based on areas, which should converge to a true minimal surface.

Permalink Reply by Florian Frank on November 8, 2012 at 1:25am

That's what most of the people don't know i think. Thanks for this clear explanation. The most critical thing is, that looking at the mesh after relaxation it looks reasonably a minimal surface but a prooving by measure the mean curvature of these surfaces it's pretty impossible, due to the mesh geometry.

That will be cool to have another method of calculation so hopefully all the guys out there, that think of having a minimal surface see that the one is not the other ;-)

Often there is no solution for a minimal surface (just a close approximation of that) but relaxation does always give a result, which i think is delusive for people who don't know.

Of course the surfaces in our project were not exact minimal surfaces, due to the transformations and morphings. And we knew that, but this project was more about connections and sequence of the spaces with a nice smooth transition between.

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