
After 400 years, mathematicians find a new class of solid shapes - srisa
http://theconversation.com/after-400-years-mathematicians-find-a-new-class-of-solid-shapes-23217
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dalek_cannes
1\. The original article: [http://theconversation.com/after-400-years-
mathematicians-fi...](http://theconversation.com/after-400-years-
mathematicians-find-a-new-class-of-solid-shapes-23217)

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 __ "

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

~~~
JoeAltmaier
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. "

~~~
JoeAltmaier
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!

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jjoonathan
> convex

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JoeAltmaier
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.

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ehartsuyker
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.

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JoeAltmaier
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.

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robinhouston
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](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!

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ttctciyf
(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-...](https://www.sciencenews.org/article/goldberg-variations-new-shapes-
molecular-cages)

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

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

~~~
robinhouston
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.

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

[https://news.ycombinator.com/item?id=7244254](https://news.ycombinator.com/item?id=7244254)

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

[https://news.ycombinator.com/item?id=7238523](https://news.ycombinator.com/item?id=7238523)

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drakaal
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...](http://match.pmf.kg.ac.rs/electronic_versions/Match59/n3/match59n3_585-594.pdf)

"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.

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gertef
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?

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staticshock
Anyone else stumble on that "nasablueshit" typo?

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

[1]
[http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress...](http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress/index.php/about/)

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Grue3
What about Johnson solids? They were enumerated only about 50 years ago.

