Here is a video of my plenary talk at this year's Symposium on Computational Geometry, on designing 3D printed mathematical art.

- Good talk!

I was curious to see what a minimal mobius ladder/ring would look like, but you didn't go that direction :)

The question at the end about how you chose which polygon was also interesting - it'd be interesting to see the same method used on a consistent (set? or even just single) vertex with a variety of polygons. I'd expect triangles and squares to be the potentially most striking. It's interesting to me that this is even a problem - one of those things that seems like it should be simple, but isn't.Jun 13, 2014 - I think that our Mobius ladder is minimal in S^3. On the thickening issue - there isn't a very clever way to choose the orientation of the polygon around the edge, so more sides seems better - otherwise we get striking shapes that aren't saying anything meaningful about the overall object.Jun 13, 2014
- I was thinking about the 'canonical thickening' problem you mentioned: How do you turn a network with infinitly thin lines into one with printable thicker lines, such that things do not get messy near the vertices. I came up with a method. A picture is shown here:

http://westy31.home.xs4all.nl/ScrapBook/CanonicalThickening.gif

You trace out all lines with a sphere in 3D. The envelope of these spheres is a surface. Once you have this surface, you can proceed to mesh it as you would any surface.

GerardJun 14, 2014 - Gerard: by "surface", do you mean a NURBS surface? How do you generate this surface in practice?Jun 14, 2014
- Henry,

So far, I only thought of how this surface is defined, not how to actually mesh it. But I intend to think about that too, since I recently started to 3D print myself.

Anyway, maybe I should explain the surface more clearly. Suppose you put a sphere on all points of a curve. The union of these spheres will be a volume. The boundary of that volume is our surface. (Intuitively, it is the ‘envelope’ of the spheres.) Alternatively, if the original curve lies on a surface, you could use cylinders instead of spheres, aligning the cylinder axis to the original surface normals. That way, the thickened lines will have flattened tops that follow the original surface.

The meshing is relatively easy around a curve whose radius of curvature is less than the radius of the thickening. You just put mesh points on circles that lie in planes perpendicular to the curve. It gets trickier if the radius of curvature gets smaller than the thickening radius. Then some points on the circles will lie inside the spheres that correspond to other points on the original curve. A laborious procedure s to check this for all mesh points, but I suspect there are nicer ways.

GerardJun 15, 2014