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from the a-straight-walk-in-a-3-dimensional-park dept.
Mathematicians Report New Discovery About the Dodecahedron:
Even though mathematicians have spent over 2,000 years dissecting the structure of the five Platonic solids—the tetrahedron, cube, octahedron, icosahedron, and dodecahedron—there's still a lot we don't know about them.
Now a trio of mathematicians has resolved one of the most basic questions about the dodecahedron.
Suppose you stand at one of the corners of a Platonic solid. Is there some straight path you could take that would eventually return you to your starting point without passing through any of the other corners? For the four Platonic solids built out of squares or equilateral triangles—the cube, tetrahedron, octahedron, and icosahedron—mathematicians recentlyfigured out that the answer is no. Any straight path starting from a corner will either hit another corner or wind around forever without returning home. But with the dodecahedron, which is formed from 12 pentagons, mathematicians didn't know what to expect.
Now Jayadev Athreya, David Aulicino and Patrick Hooper have shown that an infinite number of such paths do in fact exist on the dodecahedron. Their paper, published in May in Experimental Mathematics, shows that these paths can be divided into 31 natural families.
The solution required modern techniques and computer algorithms. "Twenty years ago, [this question] was absolutely out of reach; 10 years ago it would require an enormous effort of writing all necessary software, so only now all the factors came together," wrote , of the Institute of Mathematics of Jussieu in Paris, in an email.
The project began in 2016 when Athreya, of the University of Washington, and Aulicino, of Brooklyn College, started playing with a collection of card-stock cutouts that fold up into the Platonic solids. As they built the different solids, it occurred to Aulicino that a body of recent research on flat geometry might be just what they'd need to understand straight paths on the dodecahedron. "We were literally putting these things together," Athreya said. "So it was kind of idle exploration meets an opportunity."
Together with Hooper, of the City College of New York, the researchers figured out how to classify all the straight paths from one corner back to itself that avoid other corners.
Their analysis is "an elegant solution," said Howard Masur of the University of Chicago. "It's one of these things where I can say, without any hesitation, 'Goodness, oh, I wish I had done that!'"
YouTube videos: A New Discovery about Dodecahedrons and Yellow Brick Road and Dodecahedron (extra) - Numberphile.
Journal Reference:
Platonic Solids and High Genus Covers of Lattice Surfaces, Experimental Mathematics (DOI: 10.1080/10586458.2020.1712564)
(Score: -1, Offtopic) by Anonymous Coward on Monday September 07 2020, @03:31AM (4 children)
quantum computer cpus expected to take the form of a dodecahedron because - why not?
(Score: 4, Funny) by takyon on Monday September 07 2020, @03:45AM (3 children)
https://images.anandtech.com/doci/15820/2019-10-28%2022.53.36_678x452.jpg [anandtech.com]
[SIG] 10/28/2017: Soylent Upgrade v14 [soylentnews.org]
(Score: 3, Funny) by c0lo on Monday September 07 2020, @03:49AM (2 children)
That's a hell of thermal nightmare. Better use Celeron.
https://www.youtube.com/watch?v=aoFiw2jMy-0
(Score: 0) by Anonymous Coward on Monday September 07 2020, @01:33PM (1 child)
Celeriac, like revenge, [semiaccurate.com] are best served cold.
(Score: 2) by c0lo on Monday September 07 2020, @02:03PM
Celeriac is best served with apple™s and buck-feta cheese, all gratefully grated and enlightingly drizzled with olive oil.
https://www.youtube.com/watch?v=aoFiw2jMy-0
(Score: 2) by c0lo on Monday September 07 2020, @03:47AM
Good to know when I get to route the traces in my next dodecahedron-shaped PCB.
I hope that Kicad is paying attention. (grin)
https://www.youtube.com/watch?v=aoFiw2jMy-0
(Score: 0) by Anonymous Coward on Monday September 07 2020, @04:45AM
Fuck learning French, THIS is the GOOD SHIT.
(Score: 2, Funny) by Anonymous Coward on Monday September 07 2020, @05:08AM (2 children)
Does this help avoid rolling natural 1's in D&D? If not, what's the use?
(Score: 0) by Anonymous Coward on Monday September 07 2020, @05:17AM (1 child)
Imagine 12 D&D games being played simultaneously. And everyone rolled 1's at the exact same moment. Then this theory comes into play.
(Score: 3, Insightful) by Opportunist on Monday September 07 2020, @08:11AM
Unfortunately there are only very few and specific moments that the d12 comes into play at all, so it's unlikely to happen.
(Score: 3, Interesting) by bzipitidoo on Monday September 07 2020, @01:17PM (1 child)
I wasn't sure what they meant by a "straight path". Is it the path you get by going around the edge of the new polygon created by cutting the polyhedron in 2 with a simple straight cut? I thought they could not mean that, because the answer would be yes, for all those shapes. What they mean is a path along a face that when it reaches an edge, continues on the adjacent face at the same angle, mirrored, as it met the edge.
I had no idea this simply stated problem was unsolved until recently. But then, Number Theory and Geometry are full of such unsolved problems. The Twin Prime conjecture, the Perfect Box (is there a solid composed of 6 rectangular faces in which all the sides and face diagonals and the central diagonal are all whole number values?), the minimum surface polyhedron that can fill space (current best is a 14 sided polyhedron very similar to the dodecahedron, just replace 2 of the opposite pentagons with hexagons, and add 2 more pentagons to match the extra edges added by changing those 2 pentagons to hexagons). The 5 Platonic solids have gotten most of the attention, too. You'd think after 2000 years, there isn't anything left to prove about them.
One polyhedron I find interesting is the space-filling rhombic dodecahedron. It's basically a 3D analog of the hexagon. Wonder if that has a straight path that returns to the originating corner without meeting any other corners?
(Score: 2) by bart9h on Monday September 07 2020, @01:43PM
That probably means a straight line on the planar projection of the polyhedron.