http://www.iflscience.com/editors-blog/ ... lid-shapes

http://theconversation.com/after-400-ye ... apes-23217

https://www.sciencenews.org/article/gol ... ular-cages

There's a seed of something interesting in the original paper referred to in the article (paper by Schein & Gayed), but it's interesting to note a few things wrong with the article and the paper itself.

Quick summary: they claim to have discovered a new class of "equilateral convex polyhedra", the fourth such class apparently, after (1) Platonic solids, (2) Archimedean solids, and (3) Kepler solids. Equilateral meaning all edges are the same length.

First thing that springs to mind is: what about the Johnson solids? What about prisms and antiprisms etc.? Seems the article left out a few important additional words from the original paper. The category's full description should have been: "equilateral convex polyhedra

**with polyhedral symmetry**". Those extra words limit us to tetrahedral, octahedral and icosahedral symmetry, rulling out Johnson solids, prisms etc.

Second, we all know the Platonic and Archimedean solids, but what are these "Kepler solids"? Kepler discovered a host of polyhedra, but "Kepler solids" is not a phrase I'd heard before. You may be familiar with the Kepler-Poinsot solids, but these are nonconvex, so it can't be those. The article describes these as "including rhombic polyhedra", and the original paper expands this as "the 2 rhombic polyhedra reported by Johannes Kepler in 1611". Ah, so this is the rhombic dodecahedron and the rhombic triacontahedron.

So were these really the only 3 sets of equilateral convex polyhedra with polyhedral symmetry known before this paper? No! In fact there are infinite sets that were already known before this. Zonohedra and zonish polyhedra spring to mind. Let's look at some examples.

Here's a simple zonohedron. This is easily made in Great Stella or Stella4D. Open a dodecahedron and from the menu select "

*Poly -> Zonohedrify*" (or hit the keyboard shortcut

*Z*), accept the default settings from the options that appear, and hey presto!

That's equilateral and convex, but apparently unknown to the authors of the paper. We're not limited to just rhombic faces either. Start again with the dodecahedron, zonohedrify again, but this time change settings from "

*Zones only (create a true zonohedron)*" to "

*Add zones to existing faces (zonish polyhedron)*". Now you'll see quite a stunning

*zonish*polyhedron, with pentagons, squares, rhombi and irregular hexagons as faces.

And these sets are infinite! You can use any initial polyhedron as a seed (we used the dodecahedron above) to generate new zonohedra or zonish polyhedra, and choose to use any combination of vertices, edges or faces as a seed. You can even use another zonohedron as a seed, hence this set being infinite. If you zonohedrify the model above again (changing back to the "Zones only" option), we get this monster (still equilateral and convex!).

Be a little careful here though if you wish to stay equilateral. Sometimes zones get combined to produce some double-length edges. This would have happened above if we'd chosen to create a zonish polyhedron. And if choosing to create a zonish polyhedron, you at least need to start with another equilateral convex polyhedron. But I digress.

The article also implies that Goldberg polyhedra previously lacked planar faces. I'm not familiar with Goldberg's work, so this may well have been the case for his versions of the shapes, probably as dictated by molecular structures which care not for such things! But it is not hard to generate them with planar faces, and I've done this before without a second thought. They are the duals of geodesic spheres, again easily created in Stella. Start with an icosahedron, and from the menu choose "

*Poly -> Create Geodesic Sphere*" (keyboard shortcut

*Ctrl+G*). Enter whatever frequency you want, say 5, and hit OK. Now switch to the dual view to see a planar-faced Goldberg polyhedron.

This one however is NOT equilateral, and the faces are no doubt more distorted than those created using Schein's method would be. Something for me to improve in Stella someday! The novel thing arising from the paper for me is not that these Goldberg polyhedra can have planar faces, but that they can have all equal edge lengths. It doesn't seem too surprising though. The hexagonal arrangement gives a lot of leeway for internal face angles to change without changing edge lengths, but it's not something I'd thought of before.

I haven't read the whole paper or tried to understand the maths involved. There's something about solving to remove the dihedral angle discrepancy between edges, but equal dihedral angles doesn't equate to equal edge lengths, so I'm not sure how they achieved that part. I think the dihedral angle thing was just to keep faces planar, with something else forcing the edges all to equal length.

I think making use of duality would be an interesting approach though. By starting with geodesic spheres, duals of the Goldberg polyhedra, the problem of keeping all those hexagonal faces planar goes away. The geodesic spheres only have triangular faces, which can't help but be planar, so you can adjust their vertices to your heart's content. Then reciprocate when ready to get a Goldberg polyhedron, which is also guaranteed to have planar faces. The question though would be how to adjust the geodesic sphere so that its reciprocal would be equilateral.