Anyone else stumble on that "nasablueshit" typo?
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After 400 years, mathematicians find a new class of solid shapes
11–20 of 21 posts
Re: After 400 years, mathematicians find a new class of solid shapes
#12It’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!
(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…
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.
Re: After 400 years, mathematicians find a new class of solid shapes
#13Earlier quoted context omitted.
Indeed. A better title may be "After 400 years, a debate over a definition begins among mathematicians."
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. "
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!
Re: After 400 years, mathematicians find a new class of solid shapes
#14Earlier quoted context omitted.
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. "
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!
> convex
Re: After 400 years, mathematicians find a new class of solid shapes
#15Earlier quoted context omitted.
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!
> convex
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.
Re: After 400 years, mathematicians find a new class of solid shapes
#16Earlier quoted context omitted.
> convex
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.
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.
Re: After 400 years, mathematicians find a new class of solid shapes
#17From 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.
Re: After 400 years, mathematicians find a new class of solid shapes
#18From 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.
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?
Re: After 400 years, mathematicians find a new class of solid shapes
#19Earlier quoted context omitted.
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.
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.
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.
Re: After 400 years, mathematicians find a new class of solid shapes
#20What about Johnson solids? They were enumerated only about 50 years ago.