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

http://theconversation.com/after-400-years-mathematicians-find-a-new-class-of-solid-shapes-23217

by srisa • 12 years ago

85 19 12 years ago

theconversation.com

ColinWright• 12 years ago

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

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

felixr• 12 years ago

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

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

robinhouston2• 12 years ago

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!

ttctciyf• 12 years ago

(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...

robinhouston2• 12 years ago

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.

dalek_cannes• 12 years ago

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 "

Codhisattva• 12 years ago

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

JoeAltmaier4• 12 years ago

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. "

lotharbot• 12 years ago

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).

staticshock• 12 years ago

Anyone else stumble on that "nasablueshit" typo?

gabemart• 12 years ago

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

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

drakaal• 12 years ago

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.

gertef• 12 years ago

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?

Grue3• 12 years ago

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

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

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