11 ms·
After 400 years, mathematicians find a new class of solid shapes
- ColinWright 13y agoPerhaps 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 13y agoWell, 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
- robinhouston 13y agoIt’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!
- ttctciyf 13y 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-shapes-molecular-cages https://www.sciencenews.org/article/goldberg-variations-new-... [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.DCSupplemental http://www.pnas.org/content/suppl/2014/02/05/1310939111.DCSu...
- robinhouston 13y agoThanks 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 13y ago1. The original article: http://theconversation.com/after-400-years-mathematicians-find-a-new-class-of-solid-shapes-23217 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 13y agoIndeed. A better title may be "After 400 years, a debate over a definition begins among mathematicians."
- JoeAltmaier 13y agoI 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 13y agoHey! 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!
- jjoonathan 13y ago> convex
- JoeAltmaier 13y agoEach 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.
- staticshock 13y agoAnyone else stumble on that "nasablueshit" typo?
- gabemart 13y agoIndeed. For the curious, it should be NASA Blueshift [1] [1] http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress/index.php/about/ http://astrophysics.gsfc.nasa.gov/outreach/podcast/wordpress...
- deleted 13y ago[deleted]
- drakaal 13y agoFrom 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/match59n3_585-594.pdf 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 13y agoInteresting. 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 13y agoWhat about Johnson solids? They were enumerated only about 50 years ago.