3 ms·

What's incredible is that the most basic statement can be proven with just a line: they just show a single, simple path that returns to the vertex without touch

by blintz 6y ago

What's incredible is that the most basic statement can be proven with just a line: they just show a single, simple path that returns to the vertex without touching any other vertices. It's kind of amazing that we are proving this fact now, 2370 years after a dodecahedron was first described. The project site has cool pictures and visualizations: http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/ http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/

furyofantares 6y ago
I know they answered more than just the yes/no question but looking at that 2d diagram I find myself unable to believe a computer was needed to answer just the yes/no question.
- ncmncm 6y ago
  Agreed. This must have been known for centuries, because anybody could cut one open and draw the line, and that's the natural way to start looking at the problem. More believable is that everybody considered that one too trivial to publish. The claim that they needed sophisticated maths to prove it works is, too, obvious nonsense. Therefore, something else, not explained, was the actual point. I wonder what it was... Anyway the real work that made it worth publishing has to be the other paths.
  - sukilot 6y ago
    Drawing the line is trivial. Knowing which like to draw is hard.
    - ncmncm 6y ago
      Again, nonsense. It is obvious by inspection exactly where to draw the line, as it is obvious by inspection that it is a solution to the problem. By contrast, that there are other solutions to the problem, and of 30 kinds, is interesting and non-obvious. So, it remains a mystery what the classical problem they claimed to have solved actually was.
- MauranKilom 6y ago
  Or even with a computer, this should have been trivial to brute-force, no? I mean, you wouldn't know that it would work ahead of time, but I find it hard to believe that no one tried it? Or am I missing some deep complexity of this problem?
  - galimaufry 6y ago
    I am wondering this as well. Disclaimer I did not read the paper. On the one hand, there are infinitely many geodesics on a dodecahedron, of unbounded length, so you can't really brute force all of them. On the other hand, the actual solution only goes through each face at most once. There are infinitely many unfoldings to try, but for each unfolding there are only finitely many paths to try, and you can get the solution on one of the minimal unfoldings.
    - blintz 6y ago
      Yeah, I mean 12! is ~479 million, and there are 20 vertices, so that is very much within brute-force-on-a-laptop range. For some net, I think given a vertex + list of faces, there is a simple-to-get yes/no answer on whether a ray passing through the faces and returning to the vertex exists. I wonder if you can just write a Python script to do it... Edit: Even simpler - there are 43,380 nets of a dodecahedron, and each has 20 vertices. So if you just draw ~8 million straight lines and see if any are both on the net and do not intersect another vertex, you'll find (at least) one that works!

by blintz 6y ago

furyofantares 6y ago

I know they answered more than just the yes/no question but looking at that 2d diagram I find myself unable to believe a computer was needed to answer just the yes/no question.

ncmncm 6y ago
Agreed. This must have been known for centuries, because anybody could cut one open and draw the line, and that's the natural way to start looking at the problem. More believable is that everybody considered that one too trivial to publish. The claim that they needed sophisticated maths to prove it works is, too, obvious nonsense. Therefore, something else, not explained, was the actual point. I wonder what it was... Anyway the real work that made it worth publishing has to be the other paths.
- sukilot 6y ago
  Drawing the line is trivial. Knowing which like to draw is hard.
  - ncmncm 6y ago
    Again, nonsense. It is obvious by inspection exactly where to draw the line, as it is obvious by inspection that it is a solution to the problem. By contrast, that there are other solutions to the problem, and of 30 kinds, is interesting and non-obvious. So, it remains a mystery what the classical problem they claimed to have solved actually was.
MauranKilom 6y ago
Or even with a computer, this should have been trivial to brute-force, no? I mean, you wouldn't know that it would work ahead of time, but I find it hard to believe that no one tried it? Or am I missing some deep complexity of this problem?
- galimaufry 6y ago
  I am wondering this as well. Disclaimer I did not read the paper. On the one hand, there are infinitely many geodesics on a dodecahedron, of unbounded length, so you can't really brute force all of them. On the other hand, the actual solution only goes through each face at most once. There are infinitely many unfoldings to try, but for each unfolding there are only finitely many paths to try, and you can get the solution on one of the minimal unfoldings.
  - blintz 6y ago
    Yeah, I mean 12! is ~479 million, and there are 20 vertices, so that is very much within brute-force-on-a-laptop range. For some net, I think given a vertex + list of faces, there is a simple-to-get yes/no answer on whether a ray passing through the faces and returning to the vertex exists. I wonder if you can just write a Python script to do it... Edit: Even simpler - there are 43,380 nets of a dodecahedron, and each has 20 vertices. So if you just draw ~8 million straight lines and see if any are both on the net and do not intersect another vertex, you'll find (at least) one that works!