

Proof confirmed of 400-year-old fruit-stacking problem - andrewljohnson
http://www.newscientist.com/article/dn26041-proof-confirmed-of-400yearold-fruitstacking-problem.html#.U-pszCddXR0

======
bumbledraven
[https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion](https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion)

 _August 10, 2014_ _We are pleased to announce the completion of the Flyspeck
project, which has constructed a formal proof of the Kepler conjecture. The
Kepler conjecture asserts that no packing of congruent balls in Euclidean
3-space has density greater than the face-centered cubic packing. It is the
oldest problem in discrete geometry. The proof of the Kepler conjecture was
first obtained by Ferguson and Hales in 1998. The proof relies on about 300
pages of text and on a large number of computer calculations. [...]_

I wonder if their proof process generated a certificate that would allow a
single computer to verify their result in a reasonable amount of time?

~~~
gghh
Writing a formal proof in HOL Light is very much like writing a computer
program. If the program run from start to finish without throwing errors, the
proof is correct.

Writing a proof takes forever, because you have to take into account all
possible corner cases etc, but I'd be surprised if running (hence: verifying)
the entire Hale's proof would take more than a few minutes.

~~~
bumbledraven
_I 'd be surprised if running (hence: verifying) the entire Hale's proof would
take more than a few minutes._

I would have said the same thing before reading the announcement, but it says
it took 5000 hours for some of the computations:

 _The term the_nonlinear_inequalities is defined as a conjection of several
hundred nonlinear inequalities. The domains of these inequalities have been
partitioned to create more than 23,000 inequalities. The verification of all
nonlinear inequalities in HOL Light on the Microsoft Azure cloud took
approximately 5000 processor-hours. Almost all verifications were made in
parallel with 32 cores, hence the real time is about 5000 / 32 = 156.25 hours_

Would those computations have to be repeated by every verifier? Or is
theresome kind of certificate that can be verified faster?

~~~
murbard2
Indeed there is,
[http://en.wikipedia.org/wiki/PCP_theorem](http://en.wikipedia.org/wiki/PCP_theorem)

~~~
bumbledraven
I suppose I was unclear. I should have asked, is the file ready for me to
download? (Not, "does such a certificate exist, in the platonic sense")

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colanderman
Note that it was already known that FCC was the best _periodic_ packing (740‰
density); this was proven by Gauss. What was not known was whether there was a
more efficient _aperiodic packing_ , such as those described in
[http://en.wikipedia.org/wiki/Quasicrystal](http://en.wikipedia.org/wiki/Quasicrystal).

------
cgh
There is a good popular math book about Kepler's Conjecture: "Kepler's
Conjecture: How Some of the Greatest Minds in History Helped Solve One of the
Oldest Math Problems in the World", by George Szpiro. I enjoyed it and
recommend it if you want a historical perspective on this problem.

------
krick
I'm not saying that proof is bad or result isn't result enough, or that
computer cannot help with math research — that would be ridiculous to say. But
I strongly feel that all the news these days about results like this one are
something that could be called either "overreaction" or "misunderstanding".
Using computer to brute-force through hypothesis _might_ be useful, and works
like Voevodsky's one are both cool and useful, but when something like that
becomes overrated it seems kinda sad to me.

"Machines do the grunt work and leave humans free for deeper thinking",
really? Let me put it this way: what is the most important thing about, say,
Pythagorean theorem? From the engineer's point of view it could be ability to
find the length of the side of right triangle given the other two or something
like that. From the mathematical point of view the most important thing about
that or any other theorem is its _proof_. It isn't actually that important
that you cannot divide arbitrary angle by 3 equal parts using only compass and
the ruler — the way how you prove it is what really matters.

So, once again, autoverifiers are wonderful and so on, but if someone thinks
that writing incomprehensible, but _correct_ proofs is as fine as it gets —
it's huge mistake to think so. Math is not about constructing and computing,
it is about _understanding_ in the first place.

~~~
sillysaurus3
The four color theorem was proved by brute forcing every possible
contradiction with a computer and finding that no contradiction existed.
Therefore the four color theorem was proved to be true. Why is that a problem?

~~~
lutusp
At the time, the problem was that most reviewers couldn't understand the code
that produced the result, so they felt that they were accepting a result they
couldn't meaningfully review or independently confirm.

A lesser issue was philosophical -- it didn't seem like mathematics was
"supposed" to be conducted. This second issue has pretty much evaporated in
the intervening years.

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kenko
The image on the sidebar is captioned "Oranges are not the only stacking
fruit", which is ... interesting.

[http://en.wikipedia.org/wiki/Oranges_Are_Not_the_Only_Fruit](http://en.wikipedia.org/wiki/Oranges_Are_Not_the_Only_Fruit)

~~~
thret
Interesting spot to plug a book. Is it any good? My book list has been
shrinking the last few months.

~~~
kenko
I have no idea! I've never read it, but it has a memorable title. The weird
allusion can't have been accidental, though it is pretty strange.

------
pera
The Dense Sphere Packings book can be downloaded here:
[https://code.google.com/p/flyspeck/source/browse/trunk/keple...](https://code.google.com/p/flyspeck/source/browse/trunk/kepler_tex/DenseSpherePackings.pdf)

(for some reason the link in the flyspeck project home is dead)

------
colanderman
I was surprised that face-centered cubic was the winner; I would have thought
hexagonal close-packed would be; but the Internet tells me they have identical
density, which blows my mind.

~~~
Steuard
If you were to sit down with a pile of beads and try to build a big HCP stack
and a big FCC stack layer by layer, you'd quickly understand why the densities
are the same. The bottom two layers (call them A and B) are identical between
the two (without loss of generality), but you then have two distinct ways to
proceed.

For HCP, you shift back and add a layer identical to A (that is, the beads of
the third layer are directly above the beads of the first) and then repeat,
making an ABABAB... pattern. For FCC, you shift to the one allowed position
that is _not_ identical to A, making an ABCABC... pattern. (Fig. 1 of this
Wikipedia article is at least a little useful for visualizing this:
[http://en.wikipedia.org/wiki/Close-
packing_of_equal_spheres](http://en.wikipedia.org/wiki/Close-
packing_of_equal_spheres))

~~~
colanderman
Moreso I think what throws me is that in your example, the FCC cubes are
tilted. (i.e. if you build a FCC stack starting with a grid of squares rather
than triangles, it's much more difficult to visualize where the closely-packed
triangles lay.)

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jeffdavis
What is the definition of "efficient" in this context?

~~~
colanderman
Packing density (i.e. ratio of fruits/spheres volume to unit volume).

------
lutusp
Quote: " ... this provides a foundation which ensures _the computer can check
any series of logical statements to confirm they are true_." (emphasis added)

Not really. I wonder if the author of the linked article realizes that the
Turing Halting Problem, and Gödel's incompleteness theorems, are deeply
connected and prevent the claimed outcome as stated.

~~~
shasta
The quote pretty obviously is referring to checking a proof (series of logical
statements) not an arbitrary logical statement. Compared to most reporting on
technical subjects, this reporting was wonderful.

~~~
lutusp
> The quote pretty obviously is referring to checking a proof (series of
> logical statements) not an arbitrary logical statement.

Checking a series of logical statements, and generating a series of logical
statements, are not fundamentally different with respect to the issue of
undecidability.

> Compared to most reporting on technical subjects, this reporting was
> wonderful.

Until I got to the passage I quoted, I agreed completely, and I agree in
general in spite of it. It's one of those often-heard statements that make
more experienced readers say, "Umm, wait, you really don't want to say it that
way."

~~~
gohrt
Checking a series of logical statements is just arithmetic.

Generating a series of logical statements is only hard because the formal
statements are not generated axiomatically, they are sourced from ill-
understood human creativity. Once a formalism is _defined_ it is trivial
O(e^n) algorithm to generate all proofs of size n.

[http://math.stackexchange.com/questions/324867/automated-
pro...](http://math.stackexchange.com/questions/324867/automated-proof-
checking-machine)

~~~
colanderman
To be fair, n may be arbitrarily large, hence the undecidability of logics in
general. (Not disagreeing with you, just clarifying.)

