10 ms·

Mathematicians Report New Discovery About the Dodecahedron

172 points by theafh 6y ago

deleted 6y ago
[deleted]
WoahNoun 6y ago
For those interested in reproducible research, the authors have made their Sage notebooks available. Arxiv link to the paper: https://arxiv.org/abs/1811.04131 https://arxiv.org/abs/1811.04131 Sage notebooks: http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/ http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/
- pmiller2 6y ago
  Alternate link to paper: http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/join_closed_sc_figs.pdf http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/join...
- pmiller2 6y ago
  > reproducible research This has not been a concern in mathematics up to relatively recently, mostly because computers were generally too puny to be of much use. By "recently," I mean from the birth of the computer up through probably 15-20 years ago. Since then, there's been a real convergence of tools and computing power that's going to send mathematics in new and interesting directions over the next decades. Unfortunately, the mathematics curriculum doesn't seem to have caught up. (Please correct me if I'm wrong here -- and I'd be happy to be wrong.) If you wanted to learn these skills as a grad student even 10 years ago, you would have to go to the CS department, most likely, unless you were at one of the handful of departments doing formalized mathematics. I believe that's still the case.
  - WoahNoun 6y ago
    When I got my undergrad about a decade ago, one of the core freshman classes was a lab course on Mathematica. Not CS style programming, but it was immensely helpful in the rest of my classes for tedious calculations, eg calculating a Groebner basis or Taylor expansions of a Lie group.
  - mindcrime 6y ago
    Unfortunately, the mathematics curriculum doesn't seem to have caught up. (Please correct me if I'm wrong here -- and I'd be happy to be wrong.) If you wanted to learn these skills as a grad student even 10 years ago, you would have to go to the CS department, most likely, unless you were at one of the handful of departments doing formalized mathematics. I believe that's still the case. I would guess that by "these skills" you mean something beyond just programming in the general sense, or even numerical computation, as there shouldn't be any harm in leaning on the C.S department for that. No? Are you instead proposing that a new course be added to the mathematics curriculum, that might be titled something like "Computational Mathematics" or "Experimental Mathematics" or something along those lines? If so, what kinds of things would go in such a class in your eyes? Since then, there's been a real convergence of tools and computing power that's going to send mathematics in new and interesting directions over the next decades. Indeed. One of the things I'm interested in is learning more about Interactive Theorem Provers. I just bought a book on Coq and intend to start diving down that rabbit-hole. Interestingly, I'm only just now starting to learn to do proofs based math at all, so I'm especially (but not exclusively) curious about whether or not there's an pedagogical value in starting to learn to use these proof assistant type tools right alongside learning to do proofs. It may well turn out that the answer is "no", but it should be fun to explore all of this in either case.
    - lkozma 6y ago
      There could be more courses that teach the necessary programming techniques but also case studies of problems that have been solved this way. For example, the course "Experimental Mathematics" given by Doron Zeilberger.
      - mindcrime 6y ago
        For example, the course "Experimental Mathematics" given by Doron Zeilberger. Interesting stuff. Link, for anybody else who's curious: https://sites.math.rutgers.edu/~zeilberg/teaching.html https://sites.math.rutgers.edu/~zeilberg/teaching.html
      - ABeeSea 6y ago
        I love his blogging and writing even if I don’t agree with most of his opinions.
  - boothby 6y ago
    Counterexample here, I started working with William Stein in 2006, contributing to Sage Math. As a math student, I actually learned these skills (including source control, unit testing, peer review, etc in addition to high-performance algorithms) as an undergrad and continued to sharpen them through my grad career. True, I was seen as an oddity, but my skills were valued by colleagues and I took every opportunity I could to teach in computational summer programs.
    - mhh__ 6y ago
      You raise an important point. These skills are very important, but they are barely raised when we teach scientists how to copy and paste python snippets. In a British university it's even harder to acquire these skills because the course is often set in stone (e.g. I wrote my first - not very good! - compiler several years ago now but I still have to take Introduction to Python in effect twice) When I was 15/16, I was very lucky to have a computing teacher who - although he didn't teach us much actual programming - absolutely drilled into us the importance of a program's quality and aims. The absolute basics go a very long way: Should this function be smaller, is this logic simple etc.
  - jhoechtl 6y ago
    Is Fortran still a thing these days in a mathematics curriculum?
    - pmiller2 6y ago
      More or less no, in my experience, unless you're specializing in such a way that it would be useful. And, in that case, you're probably better off doing your degree in a CS department, anyway.
    - TheRealKing 6y ago
      yes, it is, in Physics, astronomy, chemistry, petroleum, climate and space sciences and the engineering fields.
- est31 6y ago
  Will this be archived the same way the paper is archived? Or will it be lost in 200 years like Fermat's proof of his last theorem?
  - archgoon 6y ago
    Well, Github archived it's 2020 snapshot in Svalbard. This probably won't be a one off thing, so if you're concerned; upload it to a github repo and they'll probably do it again. https://archiveprogram.github.com/ https://archiveprogram.github.com/
  - sukilot 6y ago
    There's no evidence Fermat had anything close to a proof.
    - est31 6y ago
      We have the word of a world known mathematician (Fermat). But yeah if he had written down the proof somewhere that he had in mind, maybe it would have turned out to have had mistakes in it, and it wouldn't have ever become such a large story/theme that it is now. Anyways, it's important to archive the knowledge in a way that it can be understood by future humans as well. Some random website under an .edu can disappear at any moment.
pmiller2 6y ago
> 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. Right here is the most interesting part of the article for me. It's definitely cool that we're making these sorts of discoveries still, but it's more interesting how so many more problems are now being solved via computer. Had I gone all the way through my PhD (in math), I'd be some combination of worried that my computing skills were up to snuff, and intrigued about the increasing role of computers in mathematics.
- deleted 6y ago
  [deleted]
- xxpor 6y ago
  Given most of us carry a computer with us at all times that by itself can do billions of calculations a second, I'd be more concerned/intrigued if this kind of stuff WASN'T being solved by computers these days. Even if it's just for brute forcing, that's obviously a lot more practical with computers than if we were doing stuff by hand or slide rule.
  - kazagistar 6y ago
    Given incompleteness and like the difficulty of the busy beaver problem, its pretty clear why brute forcing isn't really effective for much of math. Plus, the results it gives are maybe a bit less interesting because they don't provide as much insight on a clean and simple structure. How math works is cool and all, but why path works can open many new insights.
    - dhosek 6y ago
      On the flip side, the computer proof at least resolves the question of whether a given conjecture is true. The proof of the four color theorem by computer in 1976–89 opened the door to simplifications and clarifications of the proof in the decade following. This is not unlike a lot of the difficult proofs of the last century. The receommended path to understanding Wiles's proof of Fermat's Last Theorem begins with Yves Hellegouach's book, Invitation to the Mathematics of Fermat Wiles http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek and then Gary Cornell's Modular Forms and Fermat's Last Theorem http://www.amazon.com/exec/obidos/ASIN/0387946098/donhosek http://www.amazon.com/exec/obidos/ASIN/0387946098/donhosek which assumes a graduate student's understanding of number theory and then the reader is ready to actually read Wiles. When I was in grad school, I remember reading a comment from someone talking about looking at her roommate's materials for her PhD in English and realizing that while what she had was comprehensible to her even if she didn't necessarily understand any of it, her own materials for her math PhD were completely unintelligible to outsiders.
deleted 6y ago
[deleted]
082349872349872 6y ago
Who's going to take advantage of these families as a strategy for "Hunt The Interpolated Wumpus"? You hear a quiet rustling. There is a cold gale blowing.
- DudeInBasement 6y ago
  It is pitch black. You are likely to be eaten by a grue
  - 082349872349872 6y ago
    Grues are logic, not geometry: https://en.wikipedia.org/wiki/New_riddle_of_induction https://en.wikipedia.org/wiki/New_riddle_of_induction (I did have to check to ensure Goodman was antecedent to The Great Underground Empire) These days do we use temporal logics for these kinds of properties?
naringas 6y ago
numberphile did a video of this https://www.youtube.com/watch?v=G9_l8QASobI https://www.youtube.com/watch?v=G9_l8QASobI with Jayadev Athreya one of the main people involved
- mkl 6y ago
  This is very good. It's the same one embedded near the end of the article.
gonzus 6y ago
I knew this rang a bell: https://www.youtube.com/watch?v=G9_l8QASobI https://www.youtube.com/watch?v=G9_l8QASobI (from Jan 2020).
- deleted 6y ago
  [deleted]
deleted 6y ago
[deleted]
kazinator 6y ago
What is the exact rule here for how to follow a geodesic along a polyhedron? If we pick a vertex on a polyhedron and start marching across a face, in order for that to be a geodesic, how do we continue when we cross an edge onto an adjacent face? Is it simply that the there must be an equal and opposite angle of incidence? (Like, suppose we imagine ascending an A-frame roof on an angle, and then, passing the ridge, descend down the other side, following a path that matches the ascent such that there is a 180 degree of rotational symmetry around the intersection. Both segments of the path lie in a plane that makes a vertical cut through the roof. Is that it?)
- Koshkin 6y ago
  This sounds complicated. Couldn't one just flatten (temporarily) the two adjacent faces and draw a straight line across the edge? (I would imagine also that a geodesic crossing more than two faces or faces that just have a vertex in common might have to follow some edges or cross a vertex.)
  - kazinator 6y ago
    > Couldn't one just flatten (temporarily) the two adjacent faces and draw a straight line across the edge? You are right; that simple specification is equivalent to what I described. My brain was too temporarily foggy to see the obvious.
    - romwell 6y ago
      It is not merely simple, it contains an explanation of why it works as a definition. (Un)flattening is an isometry, it doesn't change lengths of lines. If you flatten a polyhedron and draw a straight line between two points, it is the shortest path between them (property of Euclidean space). Now fold the surface back, it remains the shortest path (and the geodesic).
      - deleted 6y ago
        [deleted]
imheretolearn 6y ago
What could be the real world applications of this discovery? On a more general note, does all research have real world applications?
- jfengel 6y ago
  There are almost certainly no real world applications. Most pure mathematical research never has real world applications. On the other hand, every once in while it gives you the foundations of e-commerce and is worth a trillion dollars. And it doesn't cost much, since all it requires is coffee and computers. So all told it's pretty cost effective. And no, we never have the faintest idea which fundamental research will turn out to be insanely profitable. Is it the case this time? No, probably not. But it cost very little to do, and maybe it enables something else.
- CliffStoll 6y ago
  Modern mathematics, like much of science, often isn't directed at real world problems. Consider cosmology, string theory, gravity waves ...
  - romwell 6y ago
    >Modern mathematics, like much of science, often isn't directed at real world problems. So is ancient mathematics. And non ancient mathematics. A lot of good mathematics is inspired or comes from real-world problems, but even more does not. At a very fundamental level, mathematics is a curious exploration of an interesting "What if...", where you get to make up whatever you want in that "...", and see where it gets you.
- romwell 6y ago
  >does all research have real world applications? The simple answer is: no. To quote Feynman: "Physics is like sex: sure, it may give some practical results, but that's not why we do it." - this applies to all research. Mathematics especially so, because it is not science, but an art form. The only difference between what people consider "good mathematics" and what isn't is what people consider beautiful, interesting, intriguing (usually, being correct is necessary for being beautiful, but not always: Fermat's Last Theorem has been on the minds of mathematicians for a very long time despite its proof being incorrect; and interesting conjectures are the driving force of this art). It's a very humane experience, this mathematics of ours.
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.

172 points by theafh 6y ago

deleted 6y ago

[deleted]

For those interested in reproducible research, the authors have made their Sage notebooks available. Arxiv link to the paper: https://arxiv.org/abs/1811.04131 https://arxiv.org/abs/1811.04131 Sage notebooks: http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/ http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/

pmiller2 6y ago
Alternate link to paper: http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/join_closed_sc_figs.pdf http://userhome.brooklyn.cuny.edu/aulicino/dodecahedron/join...
pmiller2 6y ago
> reproducible research This has not been a concern in mathematics up to relatively recently, mostly because computers were generally too puny to be of much use. By "recently," I mean from the birth of the computer up through probably 15-20 years ago. Since then, there's been a real convergence of tools and computing power that's going to send mathematics in new and interesting directions over the next decades. Unfortunately, the mathematics curriculum doesn't seem to have caught up. (Please correct me if I'm wrong here -- and I'd be happy to be wrong.) If you wanted to learn these skills as a grad student even 10 years ago, you would have to go to the CS department, most likely, unless you were at one of the handful of departments doing formalized mathematics. I believe that's still the case.
- WoahNoun 6y ago
  When I got my undergrad about a decade ago, one of the core freshman classes was a lab course on Mathematica. Not CS style programming, but it was immensely helpful in the rest of my classes for tedious calculations, eg calculating a Groebner basis or Taylor expansions of a Lie group.
- mindcrime 6y ago
  Unfortunately, the mathematics curriculum doesn't seem to have caught up. (Please correct me if I'm wrong here -- and I'd be happy to be wrong.) If you wanted to learn these skills as a grad student even 10 years ago, you would have to go to the CS department, most likely, unless you were at one of the handful of departments doing formalized mathematics. I believe that's still the case. I would guess that by "these skills" you mean something beyond just programming in the general sense, or even numerical computation, as there shouldn't be any harm in leaning on the C.S department for that. No? Are you instead proposing that a new course be added to the mathematics curriculum, that might be titled something like "Computational Mathematics" or "Experimental Mathematics" or something along those lines? If so, what kinds of things would go in such a class in your eyes? Since then, there's been a real convergence of tools and computing power that's going to send mathematics in new and interesting directions over the next decades. Indeed. One of the things I'm interested in is learning more about Interactive Theorem Provers. I just bought a book on Coq and intend to start diving down that rabbit-hole. Interestingly, I'm only just now starting to learn to do proofs based math at all, so I'm especially (but not exclusively) curious about whether or not there's an pedagogical value in starting to learn to use these proof assistant type tools right alongside learning to do proofs. It may well turn out that the answer is "no", but it should be fun to explore all of this in either case.
  - lkozma 6y ago
    There could be more courses that teach the necessary programming techniques but also case studies of problems that have been solved this way. For example, the course "Experimental Mathematics" given by Doron Zeilberger.
    - mindcrime 6y ago
      For example, the course "Experimental Mathematics" given by Doron Zeilberger. Interesting stuff. Link, for anybody else who's curious: https://sites.math.rutgers.edu/~zeilberg/teaching.html https://sites.math.rutgers.edu/~zeilberg/teaching.html
    - ABeeSea 6y ago
      I love his blogging and writing even if I don’t agree with most of his opinions.
- boothby 6y ago
  Counterexample here, I started working with William Stein in 2006, contributing to Sage Math. As a math student, I actually learned these skills (including source control, unit testing, peer review, etc in addition to high-performance algorithms) as an undergrad and continued to sharpen them through my grad career. True, I was seen as an oddity, but my skills were valued by colleagues and I took every opportunity I could to teach in computational summer programs.
  - mhh__ 6y ago
    You raise an important point. These skills are very important, but they are barely raised when we teach scientists how to copy and paste python snippets. In a British university it's even harder to acquire these skills because the course is often set in stone (e.g. I wrote my first - not very good! - compiler several years ago now but I still have to take Introduction to Python in effect twice) When I was 15/16, I was very lucky to have a computing teacher who - although he didn't teach us much actual programming - absolutely drilled into us the importance of a program's quality and aims. The absolute basics go a very long way: Should this function be smaller, is this logic simple etc.
- jhoechtl 6y ago
  Is Fortran still a thing these days in a mathematics curriculum?
  - pmiller2 6y ago
    More or less no, in my experience, unless you're specializing in such a way that it would be useful. And, in that case, you're probably better off doing your degree in a CS department, anyway.
  - TheRealKing 6y ago
    yes, it is, in Physics, astronomy, chemistry, petroleum, climate and space sciences and the engineering fields.
est31 6y ago
Will this be archived the same way the paper is archived? Or will it be lost in 200 years like Fermat's proof of his last theorem?
- archgoon 6y ago
  Well, Github archived it's 2020 snapshot in Svalbard. This probably won't be a one off thing, so if you're concerned; upload it to a github repo and they'll probably do it again. https://archiveprogram.github.com/ https://archiveprogram.github.com/
- sukilot 6y ago
  There's no evidence Fermat had anything close to a proof.
  - est31 6y ago
    We have the word of a world known mathematician (Fermat). But yeah if he had written down the proof somewhere that he had in mind, maybe it would have turned out to have had mistakes in it, and it wouldn't have ever become such a large story/theme that it is now. Anyways, it's important to archive the knowledge in a way that it can be understood by future humans as well. Some random website under an .edu can disappear at any moment.

pmiller2 6y ago

> 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. Right here is the most interesting part of the article for me. It's definitely cool that we're making these sorts of discoveries still, but it's more interesting how so many more problems are now being solved via computer. Had I gone all the way through my PhD (in math), I'd be some combination of worried that my computing skills were up to snuff, and intrigued about the increasing role of computers in mathematics.

deleted 6y ago
[deleted]
xxpor 6y ago
Given most of us carry a computer with us at all times that by itself can do billions of calculations a second, I'd be more concerned/intrigued if this kind of stuff WASN'T being solved by computers these days. Even if it's just for brute forcing, that's obviously a lot more practical with computers than if we were doing stuff by hand or slide rule.
- kazagistar 6y ago
  Given incompleteness and like the difficulty of the busy beaver problem, its pretty clear why brute forcing isn't really effective for much of math. Plus, the results it gives are maybe a bit less interesting because they don't provide as much insight on a clean and simple structure. How math works is cool and all, but why path works can open many new insights.
  - dhosek 6y ago
    On the flip side, the computer proof at least resolves the question of whether a given conjecture is true. The proof of the four color theorem by computer in 1976–89 opened the door to simplifications and clarifications of the proof in the decade following. This is not unlike a lot of the difficult proofs of the last century. The receommended path to understanding Wiles's proof of Fermat's Last Theorem begins with Yves Hellegouach's book, Invitation to the Mathematics of Fermat Wiles http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek and then Gary Cornell's Modular Forms and Fermat's Last Theorem http://www.amazon.com/exec/obidos/ASIN/0387946098/donhosek http://www.amazon.com/exec/obidos/ASIN/0387946098/donhosek which assumes a graduate student's understanding of number theory and then the reader is ready to actually read Wiles. When I was in grad school, I remember reading a comment from someone talking about looking at her roommate's materials for her PhD in English and realizing that while what she had was comprehensible to her even if she didn't necessarily understand any of it, her own materials for her math PhD were completely unintelligible to outsiders.

deleted 6y ago

[deleted]

082349872349872 6y ago

Who's going to take advantage of these families as a strategy for "Hunt The Interpolated Wumpus"? You hear a quiet rustling. There is a cold gale blowing.

DudeInBasement 6y ago
It is pitch black. You are likely to be eaten by a grue
- 082349872349872 6y ago
  Grues are logic, not geometry: https://en.wikipedia.org/wiki/New_riddle_of_induction https://en.wikipedia.org/wiki/New_riddle_of_induction (I did have to check to ensure Goodman was antecedent to The Great Underground Empire) These days do we use temporal logics for these kinds of properties?

naringas 6y ago

numberphile did a video of this https://www.youtube.com/watch?v=G9_l8QASobI https://www.youtube.com/watch?v=G9_l8QASobI with Jayadev Athreya one of the main people involved

mkl 6y ago
This is very good. It's the same one embedded near the end of the article.

gonzus 6y ago

I knew this rang a bell: https://www.youtube.com/watch?v=G9_l8QASobI https://www.youtube.com/watch?v=G9_l8QASobI (from Jan 2020).

deleted 6y ago
[deleted]

deleted 6y ago

[deleted]

kazinator 6y ago

What is the exact rule here for how to follow a geodesic along a polyhedron? If we pick a vertex on a polyhedron and start marching across a face, in order for that to be a geodesic, how do we continue when we cross an edge onto an adjacent face? Is it simply that the there must be an equal and opposite angle of incidence? (Like, suppose we imagine ascending an A-frame roof on an angle, and then, passing the ridge, descend down the other side, following a path that matches the ascent such that there is a 180 degree of rotational symmetry around the intersection. Both segments of the path lie in a plane that makes a vertical cut through the roof. Is that it?)

Koshkin 6y ago
This sounds complicated. Couldn't one just flatten (temporarily) the two adjacent faces and draw a straight line across the edge? (I would imagine also that a geodesic crossing more than two faces or faces that just have a vertex in common might have to follow some edges or cross a vertex.)
- kazinator 6y ago
  > Couldn't one just flatten (temporarily) the two adjacent faces and draw a straight line across the edge? You are right; that simple specification is equivalent to what I described. My brain was too temporarily foggy to see the obvious.
  - romwell 6y ago
    It is not merely simple, it contains an explanation of why it works as a definition. (Un)flattening is an isometry, it doesn't change lengths of lines. If you flatten a polyhedron and draw a straight line between two points, it is the shortest path between them (property of Euclidean space). Now fold the surface back, it remains the shortest path (and the geodesic).
    - deleted 6y ago
      [deleted]

imheretolearn 6y ago

What could be the real world applications of this discovery? On a more general note, does all research have real world applications?

jfengel 6y ago
There are almost certainly no real world applications. Most pure mathematical research never has real world applications. On the other hand, every once in while it gives you the foundations of e-commerce and is worth a trillion dollars. And it doesn't cost much, since all it requires is coffee and computers. So all told it's pretty cost effective. And no, we never have the faintest idea which fundamental research will turn out to be insanely profitable. Is it the case this time? No, probably not. But it cost very little to do, and maybe it enables something else.
CliffStoll 6y ago
Modern mathematics, like much of science, often isn't directed at real world problems. Consider cosmology, string theory, gravity waves ...
- romwell 6y ago
  >Modern mathematics, like much of science, often isn't directed at real world problems. So is ancient mathematics. And non ancient mathematics. A lot of good mathematics is inspired or comes from real-world problems, but even more does not. At a very fundamental level, mathematics is a curious exploration of an interesting "What if...", where you get to make up whatever you want in that "...", and see where it gets you.
romwell 6y ago
>does all research have real world applications? The simple answer is: no. To quote Feynman: "Physics is like sex: sure, it may give some practical results, but that's not why we do it." - this applies to all research. Mathematics especially so, because it is not science, but an art form. The only difference between what people consider "good mathematics" and what isn't is what people consider beautiful, interesting, intriguing (usually, being correct is necessary for being beautiful, but not always: Fermat's Last Theorem has been on the minds of mathematicians for a very long time despite its proof being incorrect; and interesting conjectures are the driving force of this art). It's a very humane experience, this mathematics of ours.

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.