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 "
"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. "
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).
The claim that Goldberg polyhedra are not really polyhedra is especially puzzling. Presumably the paper explains this better!
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...
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.
Is the "Extended Goldberg polyhedra" prior publication of the same result as today's news?
[1] http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress...
