
Mathematicians Report New Discovery About the Dodecahedron - theafh
https://www.quantamagazine.org/mathematicians-report-new-discovery-about-the-dodecahedron-20200831/
======
pmiller2
> 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.

~~~
coliveira
The fact that these mathematical problems can be solved by computer doesn't
mean that any computer scientist or teenage hacker can go around solving math
problems. The solution to these problems is more a matter of mathematical
skill combined with some programming than a purely programming effort. So I
don't think that mathematicians have any reason to fear for their jobs, they
just have another tool to learn to be good at what they do.

~~~
kmill
I think of computers as being telescopes for math. They can help you see
deeper than you could before, but you still need to know where to look, make
sure you have enough resolving power, etc.

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WoahNoun
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
> 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.

~~~
jhoechtl
Is Fortran still a thing these days in a mathematics curriculum?

~~~
pmiller2
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.

------
blintz
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
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.

~~~
MauranKilom
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
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
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!

------
kazinator
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
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
> _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
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).

~~~
Someone
The _“Now fold the surface back, it remains the shortest path”_ part isn’t
true, in general. A different unfolding of the polyhedron may yield a shorter
path. See
[https://en.wikipedia.org/wiki/The_spider_and_the_fly_problem](https://en.wikipedia.org/wiki/The_spider_and_the_fly_problem)

~~~
kazinator
Even if we are discussing a pair of points on adjacent polygons? Wouldn't the
shortest path go across the one and only edge the polygons have in common?

I'm looking at the diagrams on that Wikipedia page. It looks as if we require
the path to visit certain polygons, then it is in fact the shorted path which
visits those polygons.

In both solutions, the path traced along the unfolded jacket of the box is in
a straight line, obeying the principle we have been discussing.

I don't think that what we are looking at there are simply two different
"different unfoldings" of a polyhedron; they are different choices about which
configuration of edges to separate in forming what is evidently called the net
of the polyhedron:

[https://en.wikipedia.org/wiki/Net_(polyhedron)](https://en.wikipedia.org/wiki/Net_\(polyhedron\))

They are different "nettings", if you will. We can crack a polygon into more
than one net, but a given net has just one unfolding.

(If we choose a particular net of a polyhedron, can _that_ have more than one
shortest path between some two points?)

~~~
Someone
_”Even if we are discussing a pair of points on adjacent polygons?”_

Yes. Consider a pointy triangular wedge of two 1 by 100 rectangles and a third
one that’s 1 by 1, and pick points on the two large rectangles that are close
to the edge they share with the small one. The shortest path will visit all
three rectangles.

Also: I was only refuting the claim that such a line is the shortest line
between the two points. I happened to know an (IMO) interesting
counterexample, so I mentioned it. I think the conclusion that such a line is
a geodesic is valid, but not because of the argument given.

------
naringas
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
This is very good. It's the same one embedded near the end of the article.

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gonzus
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).

------
082349872349872
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
It is pitch black. You are likely to be eaten by a grue

~~~
082349872349872
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?

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

~~~
romwell
>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.

~~~
ncmncm
New vistas of mathematics open up when what was once ugly comes to be
appreciated, usually two or more generations after its first exploration.

The history shows up in terminology: "irrational" numbers were once anathema,
but had to be accepted (although the feelings about the word might have arisen
in the other direction). "Negative" numbers came a lot later, and only after
zero, which was disliked. "Imaginary" is another. After the bad numbers had to
be accepted, we needed a name for the good ones: thus, "natural".
Transcendentals got a free pass because of pi, so got a nice name.

Infinities are still causing trouble. Like irrationals, they are named for
what they are not.

