Nanotube tiling

 Hi

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For a  while I've been making some sphere like things from cane, and these are actually twin edged platonic solids, and this draft article from this post about a year ago started to classify them. I've been learning more about them since, and in the last post made item 3.3 from the table above, a dodecahedron. I have also built 1.3, 3.3 (tetrahedron and cube and am now moving on to a version of 4.3.

2. Stylised / cutout forms of parallel face shapes.

So having built all the solid "platonic solid shapes" in the series (1.3. 2.3, 3.3) how about the planar one, 4.3? Its possibly the most interesting of all because in its rolled form (ie opposite edges of pic 4.3 are joined to form a circle)  it is the shape of a carbon nanotube. The wikipedia page and this article show the various forms of carbon nanotube, which are called "Armchair" and "Zigzag" and are flavoured by spirals. So before launching into making cane models, I decided to make 3d printed ones.  These will help me design and build the jigs for the cane versions.

The designs shown below can be made by 3d printing using the "hex" thingiverse file available here. (The file is a recent addition to the 3d printable construction set). First parts were printed, then they were assembled in sheets using orange 3d printer filament as pins between parts. Finally, sheets were rolled and edges were joined, again using 3d printer filament. Sheets could be Zigzag or Armchair. Plain joining created cylinders while displaced joining created spirals.

All the rolled parts have a resistance to rolling created by conflicting surfaces within the structures. This is something already observed and explained in non-hexagonal cylinders, see the pdf available on the thingiverse page.

3. Plain zigzag (style nanotube model)

4. Compatibility of hexagon with other pieces from the Thingiverse set. Parts at the top are the small equisided triangle.

5. Zigzag with spiral

6. Zigzag before rolling. (the top edge is zigzag)

7. Armchair before rolling (the top edge is armchair)

8. Plain Armchair

9. Armchair with spiral

Update July 6

10. Cad model for central jig for cane model nanotube. Aiming to print this within a few days. It is....

11 derived from the red / central jig seen here being used to make an octohedron. The shape is 5.3 in the table shown at the top

I've now designed a jig to make a cane model of a nanotube. The tiling nanotube pics on this page have helped me work out what to build and how to build it. I will start with the zigzag cylinder as modelled in 3. More blog posts to follow.

Now that I've done one hex shape, I can see the sense in doing another hex and 2 pentagons to fill in the gaps in the construction kit. When time allows!

PS: as a reference for the Nanotubes, here are 2 pages from Noel Graham's excellent book "Mathematics with Geoshapes", published in 1998.

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