(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.
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.
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...