After 400 years, mathematicians find a new class of solid shapes | Hacker News Reader

After 400 years, mathematicians find a new class of solid shapes

theconversation.com

85 pointssrisa19 comments

1. The original article: http://theconversation.com/after-400-years-mathematicians-fi...

2. It actually looks more like a redefinition than a new discovery: "It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces "

Indeed. A better title may be "After 400 years, a debate over a definition begins among mathematicians."

I don't think that's quite right. They narrowed the definition to strict polyhedral, which hadn't been done before. Then showed that they existed.

"Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others. "

Hey! There are in fact infinite solution. Each regular face of an icosahedron for instance can be 'inflated' to form a slight dome, made out of smaller regular polygons.

Then, recurse!

> convex

Each surface polygon is flat. They can be 'inflated' via the OPs technique without violating the bound of an enclosing sphere, right? Each recursive expansion has an inflation factor that scales. Hm. But the sphereical section bounding each polygon doesn't scale, it becomes 'flatter' as you recurse. So there's a limit.

Actually, not. The definition of convex is that given a point A and a point B and a line between A and B, all points on the line AB are in the interior space of the solid.

Inflating two adjacent surfaces creates a valley along the pre-existing edge between the two of them and fails the above definition.

Yet that's what the OP describe. Remember, the edge was a "mountain" to begin with, you have some wiggle room. That's the observation that the whole paper is based upon.

Goldberg polyhedra: http://en.wikipedia.org/wiki/Goldberg_polyhedron

As far as I can tell, the discovery here (if there is one at all) is a method for constructing those polyhedra and others like them and being sure they're actually polyhedral (no curved or bent faces).

It’s hard to know whether or not this is interesting, since the article is very vague and the paper is behind a paywall: http://www.pnas.org/content/early/2014/02/04/1310939111

The claim that Goldberg polyhedra are not really polyhedra is especially puzzling. Presumably the paper explains this better!

(Non-mathematician here.) Seems the fuss is about getting the faces of the Goldberg polyhedra to be planar.

There's an article at sciencenews.org [1] which has a bit better explanation I think.

It seems "Goldberg polyhedra" as commonly understood encompasses a bunch of shapes which wouldn't normally qualify as polyhedra because some of their faces don't have all of their vertices in the same plane (i.e. the "hexagons" in the picture at the article would not really be flat)

This is what the paper is calling "dihedral angle discrepancy" - a dihedral angle being the angle between two planes.

From the abstract[1], the claim of the paper is to have found a subset of Goldberg "polyhedra" where the planarity of faces is guaranteed.

The resulting shapes also have all edges the same length, but the faces are not necessarily equiangular.

As far as I can tell, they're claiming that only one each of tetrahedral and octahedral Goldberg (or Goldberg-like?) polyhedra exhibits equal edges and planar faces, but that there are infinite icosahedral variations with these properties.

The supplementary info for this paper[2] has more details about their methodology, which seems to included use of molecular modelling software and iterative methods, as well as a few pictures.

[1] https://www.sciencenews.org/article/goldberg-variations-new-...

[2] http://www.pnas.org/content/early/2014/02/04/1310939111

[3] http://www.pnas.org/content/suppl/2014/02/05/1310939111.DCSu...

Thanks for the link to the sciencenews piece. That’s much more helpful. So it’s a new class of equilateral convex polyhedra with icosahedral symmetry, which is interesting because the familiar “geodesic dome” polyhedra are not equilateral.

Perhaps this submission will get more love than when I submitted it 2 days ago:

https://news.ycombinator.com/item?id=7244254

Well, I submitted this news 4 days ago and my post also did not receive any love ...

https://news.ycombinator.com/item?id=7238523

From 2007, this is a better article on the same topic. Sorry it is a PDF, it wasn't easy to find an online version.

http://match.pmf.kg.ac.rs/electronic_versions/Match59/n3/mat...

"Our results show that these Extended Goldberg polyhedra are a kind of novel geometrical objects of icosahedral symmetry and are considered to explain some viral capsids. "

Which is the interesting application of the math.

Interesting. The "Extended Goldberg polyhedra" paper doesn't make explicit whether they are talking about planar ("proper") polyhedra, but maybe they are...?

Is the "Extended Goldberg polyhedra" prior publication of the same result as today's news?

Anyone else stumble on that "nasablueshit" typo?

Indeed. For the curious, it should be NASA Blueshift [1]

[1] http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress...

What about Johnson solids? They were enumerated only about 50 years ago.

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