Mathematicians Finally Answer 2,000-Year-Old Question About Dodecahedrons (quantamagazine.org) 31
NCamero (Slashdot reader #35,481) brings some news from the world of 12-sided dodecahedrons:
Quanta magazine reports that a trio of mathematicians has resolved one of the most basic questions about the dodecahedron. The cube, tetrahedron, octahedron and icosahedron cannot have a straight path you could take [starting from a corner] that would eventually return you to your starting point without passing through any of the other corners. The dodecahedron can.
Mathematicians studied dodecahedrons for over 2,000 years without solving the problem, reports Quanta magazine. But 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 Anton Zorich, of the Institute of Mathematics of Jussieu in Paris, in an email.
Mathematicians studied dodecahedrons for over 2,000 years without solving the problem, reports Quanta magazine. But 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 Anton Zorich, of the Institute of Mathematics of Jussieu in Paris, in an email.
Well-written article (Score:4, Interesting)
I almost understood maybe 25% of it. Considering I know next to nothing about math, that says a lot.
Re: (Score:1)
What I'm wondering, though, is why nobody just attacked it with brute force. Draw a random line, calculate with high numerical precision where it hits the next edge, transform it to an edge and corresponding direction on the next face, repeat for a while, and check how far you get from a copy of the same corner. Now shift the starting direction slightly, and see how much the distance gets closer or further. That's your derivative for Newton's root finding method. Some solutions may end up going through anot
Another interesting topic is regular polyhedrea (Score:2)
On a related note Jan Misali has a really interesting video talking about Platonic Solids and how they are related to the 48 regular polyhedra [youtube.com]. That is, while there are only 5 Platonic Solids -- which are a subset of regular polyhedra -- there are significantly more regular polyhedra, specifically 48, that tends to be ignored.
Jan covers:
* Kepler Solids
* Kepler-Poinsot polyhedra
* Petrie-Coxeter polyhedra
* Regular Skew apeirohedron
* Petrials
* Apeirohedron
* Gunbaum-Dress polyhedra
Numberphile (Score:5, Informative)
12-sided dodecahedron? (Score:3)
12-sided dodecahedron (dodeca = 12, hedra = side), as opposed to... not 12-sided dodecahedra? Even when the post is actually nerdy, slashdot editors still manage to throw some ignorance in...
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It's awkward, but it's not an altogether unusual construction to use an adjective as a description of the class of noun rather than as particularizing the noun. Hence: "The near-blind cyclops envied his twelve-eyed dodecacyclops neighbor."
Eh, not sure what you are on about, first, "dodecacyclops" is a bit absurd, as cyclops means "circle-eye". Saying 12-sided dodecahedron is the same as saying 3-sided triangle. Or, if you prefer, writing 1-eyed cyclops, on a forum about ancient greek mythology...
Of course somebody might interject that "1-eyed cyclops" is valid, because Polyphemus became a "0-eyed cyclops" after Odysseus was done with him, but that's a bit of a stretch / too pedandic.
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Haha, yes, dodecacyclops doesn't make sense etymologically. But that's how language works; take for example quadcopter. It's a portmanteau of quad + helicopter. Helicopter is formed from helico + pter, so if language made sense it'd be quadpter, or better quadripter. But that didn't happen.
Anyway, "1-eyed cyclops" is probably a better example. You can readily find such language. A National Geographic article [nationalgeographic.com] says: "To the ancient Greeks, Deinotheriumskulls could well be the foundation for their tales of the
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Haha, yes, dodecacyclops doesn't make sense etymologically. But that's how language works; take for example quadcopter. It's a portmanteau of quad + helicopter. Helicopter is formed from helico + pter, so if language made sense it'd be quadpter, or better quadripter. But that didn't happen.
Anyway, "1-eyed cyclops" is probably a better example. You can readily find such language. A National Geographic article [nationalgeographic.com] says: "To the ancient Greeks, Deinotheriumskulls could well be the foundation for their tales of the fearsome one-eyed Cyclops." It's redundant, maybe useless, but it's not incorrect grammar.
You missed my point though, I said "1-eyed cyclops, *on a forum about ancient greek mythology*". So, the submitter had not added any explanation to the dodecahedron, as he knew the audience. The slashdot editor on the other hand thought he would "helpfully" add something as redundant as "dual barrel binocular" and about as correct as "PIN number" etc, possibly because he is an idiot himself and had to look dodecahedron up.
I am confused (Score:3)
" cannot have a straight path you could take [starting from a corner] that would eventually return you to your starting point without passing through any of the other corners. "
How can a straight line (in 3D space) return you to your starting point anyway?
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A straight path, not a straight line. So if you put your pen in one corner and then rotate the object in space, can you get back to the point you started without crossing another corner.
Re: I am confused (Score:2)
What?
Re: I am confused (Score:2)
I'm sorry, what?
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Re:I am confused (Score:4, Informative)
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It's a straight path on the surface of the polyhedron (which is embedded in 3d space). Instead of starting at a corner and flying in a direction forever, it's starting at a corner and then walking on the polyhedron without changing your direction. If it helps to visualize what it means to not change your direction when crossing an edge, just imagine uncreasing that edge and walking along that flat surface, then re-folding it once you've crossed it.
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Try flattening the faces of a cube, gluing a bunch of copies of this flattening together, and then connecting a corner to a copy of itself. You won't be able to without crossing another corner. Just think of the classic "T" flattening, and start at one of the points at the bottom of that "T". You would need to connect it to, for example, the point at the top immediately above it. That line obviously goes through a bunch of corners. So you try something different. No matter how you arrange the multiple copi
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The idea of flattening and refolding at an edge answers one of my questions about what it means to travel in a straight line over the surface of a polyhedron. I think it is a generalisation of geodesics when traveling long distances over the earth. One surpising effect of the 3D geometry is that, if you want the shortest air route between London and Los Angeles, you aim north, and travel over the southern tip of Greenland. Radio amateurs are aware of this. In the UK, signals from the US are likely to come f
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> How can a straight line (in 3D space) return you to your starting point anyway?
Easy. IF 3D space is non-Euclidian -- i.e. a torus -- then you would "wrap around". (Note: The shape of the universe [wikipedia.org] is currently an unsolved problem. Astrophysicist Alexei Starobinsky proposed the Three-torus model of the universe [wikipedia.org] back in 1984, etc.)
However, for THIS problem:
a) The phrase "straight line" is misleading as there are multiple lines.
b) We're talking about straight lines restricted to ALWAYS being on the surfac
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This was my first reaction. If a line has to cross an edge between two faces, then surely the line is not straight. What determines the angle the line makes across the new face?
The idea of a straight line returning you to the same place makes some sense. For example, for a being living in Flatworld, on the surface of a 3D body, it is quite possible to set out in a straight line and end up where you started, without ever changing course.
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" cannot have a straight path you could take [starting from a corner] that would eventually return you to your starting point without passing through any of the other corners. "
How can a straight line (in 3D space) return you to your starting point anyway?
I didn't understand either until I clicked the link. The picture at the top makes it pretty clear.
I’m more interested to... (Score:4, Funny)
Learn about 3 sided dodecahedrons, as that would be more impressive.
This is BIG (Score:2)
The practical implementations of this discovery are going to be huge!
Interesting (Score:2)
There are many facets to this problem. Twelve, in fact.
this summary may contain FALSE information (Score:2)
author says "an infinite number of such paths do in fact exist" on a dodecahedron, and links to an article in the Journal of Experimental Mathematics.
have only read the abstract, however it does not claim to find an infinite number of paths, and an infinite number of paths is plainly not possible on a finite 'hedron.
you do not need a lot of maths to see this, as far as i know.
am i wrong?
oops, yes, i am wrong (Score:2)
read the premise wrong, got it mixed with a simpler problem, way less smart than i think i am.