
Mathematicians find way to put 7 cylinders in contact without using their ends - ColinWright
https://www.sciencenews.org/article/tale-touching-tubes
======
ColinWright
It has been known for a long time that one can arrange 7 cylinders to be
mutually touching. That was written about by Martin Gardner decades ago, and
was set as a puzzle.

The result had the cylinders touching at the end of one with the length of the
other, so the question arose, can one arrange to have seven cylinders all
mutually touching, without using the ends. The easiest way to say this is to
ask for seven infinitely long cylinders mutually touching.

This has only recently been settled, hence this paper. It's believed
impossible to arrange eight identical infinitely long cylinders to be mutually
touching. I suspect the result is in fact known, but I haven't searched
diligently for it.

There is an associated puzzle that uses cylinders that are very short - think
coins. How many coins can you arrange to be mutually touching?

Consider that a puzzle. I can do 5. If you can do more, there's a mathematical
paper in it for you, should you care.

~~~
lotharbot
I linked to another related puzzle elsewhere in this discussion: there appears
to be a solution for 9 cylinders of infinite length but _different radii_.
It's not clear what's known about the case of 10 cylinders.

So there are known solutions for 5 coins, 7 identical infinite cylinders, 7
identical finite cylinders (maybe more), and 9 different infinite cylinders.

~~~
darkmighty
Is there an example of a similiar non-trivial problem for which there is an
inexistance proof? I'm not a mathematician so I have a hard time picturing a
way of proving this sort of thing when the number of objects is "tricky" (not
too high, not too low), apart from simply showing a counterexample.

~~~
Strilanc
How about the fact that there are no more than 5 platonic solids [1]? (See the
proof in the article)

1:
[http://en.wikipedia.org/wiki/Platonic_solid](http://en.wikipedia.org/wiki/Platonic_solid)

------
ColinWright
Question I've not found the answer to:

    
    
        It's possible to have 7 arbitrarily long cylinders
        mutually touching.  Currently it's not possible to
        have more than 5 coins (which are short cylinders)
        mutually touching.  As the cylinder's aspect ratio
        decreases, where are the thresholds: 7 -> 6 -> 5 ?

------
chubot
This sort of reminds me of:
[http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c](http://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c)

Because it is a 3D object that was found using mathematics. Any other
examples?

I think there are lots of new objects discovered in higher dimensions, but I
like when there is something you can actually build and see. I also like how
it appears to be very asymmetrical.

~~~
pavelrub
The Klein bottle is sort of like that, only it's supposed to be in 4
dimensions, but the 3d version is still pretty cool and quite famous!

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

Never heard about the gomboc before, thanks!

~~~
anigbrowl
Clifford Stoll of _Cuckoo 's Egg_ fame has a side business making glass Klein
bottles. He lives near me, I sometimes see him at the post office shipping out
a bunch of them.

~~~
Someone
His site has a cool domain name:
[http://www.kleinbottle.com](http://www.kleinbottle.com)

------
josephwegner
This is going to sound like trolling, but it's not - I'm honestly curious.

Why is this important? Is it just cool, or is there some real world
application? Was someone paying for this research for some reason, or was it
just a mathematician's hobby?

EDIT: For the record, I don't have any problem with "just cool" research. I do
that kind of research often (albeit, not as smart), and totally understand the
value in it. Just wondering if this had an application immediately.

~~~
wesleyd
I believe this result has important applications in cylindrical stack
encryption; it's very difficult to produce a solution, but very easy to verify
a given solution.

~~~
jafaku
Brb, I'm gonna create CylinderCoin.

~~~
pjbrunet
I'd buy some. Would be awesome if each (coin) solution was rendered with a
raytracer.

------
soneca
Great design for a no-gravity space station.

Easy to go everywhere from everywhere.

~~~
aaronem
I was going to make a vaguely snide comment about difficulty in routing and
managing services (air, water, power, data, &c.), but the more I look at the
model, the less firmly I'm convinced that there is an intractable topological
problem involved.

I'm still going to go with calling it the Villa Straylight, though.

~~~
gatehouse
You could dedicate a cylinder to utilities, and it will connect directly to
all the others.

~~~
aaronem
That was more or less what I had in mind. On the other hand, it might be more
cost-efficient in general to design the cylinders to connect endwise and form
a hexagon; it'd increase travel time for the station's inhabitants, but reduce
the need for unique construction tooling -- instead of seven types of
cylinders, each of whose interconnections are placed differently from those of
all the others, you build one type of cylinder and one type of connector, then
fit out each unit according to the purpose it serves in a given station
design. You can also enable more complex topologies simply by developing more
complex connectors, whereas what I shall please myself by calling the
"eccentric" design under discussion doesn't scale nearly as well.

~~~
pjbrunet
Or you could forego connectors and have them all intersect slightly. Once in
space, pop out the panels where the intersections will happen.

~~~
aaronem
That scales better than the eccentric design, but not as well as separate
connectors.

Suppose, for example, you want to be able to assemble, from the same
components, a flat-hexagon station, and an "asterisk" station like what you'd
get if you drew radii from each vertex of a hexagon and then took the hexagon
away -- a configuration which might be very useful, for example, as a
"transfer point"; if the inter-segment connectors are adaptable to whatever
standard governs spacecraft airlocks, then you can dock at least six
spacecraft at once and interchange cargo among any or all of them. If you add
a seventh "socket" to your central connector, normal to the plane of the other
six, then you can tie your "transfer point" to a larger station, too, and
assemble a larger structure, as for example might be very useful at a nearby
Lagrange point, as a way station for ships inbound to and outbound from the
Earth-Moon system.

(On the other hand, perhaps I simply spent too much time playing with
Tinkertoy as a child.)

------
huhtenberg
Here's the original solution -
[http://www.mathpuzzle.com/7cylinders.gif](http://www.mathpuzzle.com/7cylinders.gif)

------
analog31
Ask HN: Is there an explanation somewhere of this "certification" that a
layperson (college math and physics major, and self taught programmer) could
understand?

~~~
darkmighty
The article is quite readable. I'm a layperson too, but here's what I made of
the methods provided:

1- The first method finds a constant that guarantees convergence of newtons
method to the solution; they then iterate it a few times and verify the result
is almost stationary.

2- They use a set with parameter r that is guaranteed to contain a solution if
it's contained in a ball of radius r; they then show that a given small ball
around the candidate solution contains this set (they were able to do that
because this set gets smaller faster than the radius).

------
DonGateley
So, given that this involves rounding errors in solutions to equations it is
only an approximate solution. A solution with fuzz. Is it possible to prove
that each point of "contact" is exactly coincident? Or to prove that exact
coincidence is not possible. There seems to be room for deeper work on this
problem.

~~~
ColinWright
The initial space for the solution was found with techniques that had the
possibility of rounding errors, and then exact proofs were found. This is a
standard technique, one that I used in my PhD. So yes, it is possible to prove
that each point of "contact" is exactly coincident, and if you read the paper
carefully you'll see that that's what the authors did.

------
josh-wrale
Manufacturing errors? I can't tell if this is plastic, but if it is, surely a
machinist can do better with metal.

~~~
delinka
By "errors" they mean "the minute imperfections created by molding, machining,
carving, etc." The rounding errors in the computer model would produce smaller
physical artifacts than the manufacturing method would.

~~~
tehwalrus
Surely mathematicians would use rationals[1] rather than floating point
numbers, thus eliminating rounding errors at the expense of performance?

[1] [https://gmplib.org/](https://gmplib.org/)

~~~
lifthrasiir
It is not a matter of rational numbers, since the solution is likely to be an
irrational number.

The paper [1] describes a system of 20 polynomial equations with 20 variables
(Equations 10--12) and solving them is no trivial task. To the end, authors
first numerically found candidate solutions up to some ten decimal digits [2]
and used specialized tools to prove (!) that there exists a real solution
sufficiently close to given numerical solution. It turns out that there are
several ways (and corresponding implementations) to do that, e.g.
alphaCertified [3].

[1] [http://arxiv.org/abs/1308.5164](http://arxiv.org/abs/1308.5164)

[2] 10^-11 to be exact, according to the paper. It would be much easier to
find a near-solution with larger errors (say, 10^-5). The physical model of
such near-solution would be indistinguishable to the correct model.

[3]
[http://www.math.tamu.edu/~sottile/research/stories/alphaCert...](http://www.math.tamu.edu/~sottile/research/stories/alphaCertified/)
(that actually does use GMP and MPFR for the obvious reason)

~~~
tehwalrus
Interesting. I would have thought that one could solve such things exactly by
representing each unique known irrational that arises (root 2, pi, etc) by its
own rational multiplier, and then overloading the relevant equality checks. Of
course, you'd need to anticipate/implement each irrational type that might
arise (roots, the geometric transcendental pi, and so on.)

[Leaving the next sentence in, for comedy value. I typed it and then realised
how ridiculous it is - apologies!]

Essentially it's just teaching the computer to do the algebra for you, isn't
it?

~~~
lmm
How would you test whether two generic irrational numbers were equal?
Obviously you can numerically approximate them and if you see any difference
in the numerical approximation then they must be different - but if they seem
the same up to e.g. 10 decimal places, what do you do next?

~~~
tehwalrus
I would generally assume that sqrts of primes don't overlap, so you can always
do exact comparisons of rational coefficients? It wouldn't be perfect, but it
would do as well as a person with pen and paper and a hundred years.

[[ Approximate algorithm, in case I'm not being clear: you need sqrt(35),
represent it as 1×sqrt(5) × 1×sqrt(7). You simplify each expression evaluated
down to roughly what you'd write on paper in RAM, and then you do exact
comparisons - is that the same coefficients of the same number of the same
prime sqrts? For greater than/less than you cast them to a hundred-sig-fig
float, and if those are still equal, keep going down the rabbit hole.
Obviously the RAM requirements for complicated numbers would be large, but
that's the same with rationals - this is just taking it to the extreme. I'd be
very surprised if some function of sqrts was _exactly the same as_ some other
function of different sqrts. ]]

~~~
lmm
Sines of certain fractions of pi are equal to certain ratios of square roots
though. And once you get to more complicated functions it's very hard to prove
anything (e.g. I believe it's still not even proven that pi^e is irrational,
never mind transcendental). Could we construct a countable field (for to be
able to write down the numbers involved the field would have to be countable,
i.e. only an infinitesimal fraction of the reals) that contained everything we
need to do this kind of geometry in? Maybe. But I can just as easily believe
you could construct a way of arranging objects that required a turing-complete
computation to determine whether two objects met, in which case the halting
problem makes this impossible.

------
kang
Links from one of the best riddle website :
[http://www.wuriddles.com/cigarettes.shtml](http://www.wuriddles.com/cigarettes.shtml)

Cylinder Length | Max cylinders that can touch | Min cylinders that can touch

Infinite | 7 | 5

Actual | 9 | 7

L=D | 4 | 4

~~~
ColinWright
Those "max" values are speculation, while the "Min" values are for definite
because there are constructions. So that table is of less value than you might
think.

------
thret
This is just one of many puzzles and mathematical curiosities popularized by
Martin Gardner. His books and columns make delightful light reading for anyone
with a curious mind.

------
laxatives
Never seen this sort of thing before. Does this hold for an arbitrary radius?
What if these were just lines in 3 space?

~~~
acchow
Intuitively, it seems that radius/length ratio is what's important here.

~~~
pisarzp
No, as this cylinders can be infinitely long if I'm not mistaken

~~~
acchow
Sure, there's no lower bound on the radius/length ratio.

But there is an upper bound (think huge radius, short length).

~~~
ColinWright
Yes - currently the best result for coins is 5 mutually touching. Neither
beaten, nor proven to be maximal.

------
Zitrax
Something special about 7 cylinders or would this be equally hard/simple for 6
or 8 ?

~~~
SamReidHughes
It gets harder and harder (and then impossible) the more cylinders you have.

~~~
cousin_it
Now I wonder if there's a proof that it's impossible for 8...

~~~
lotharbot
There's an arrangement of 8 that's very close to touching, but has been proven
to not actually touch:
[http://www.sciencedirect.com/science/article/pii/S0195669808...](http://www.sciencedirect.com/science/article/pii/S019566980800022X)

Others believe they have found both an 8 and a 9:
[http://arxiv.org/pdf/1312.6207.pdf](http://arxiv.org/pdf/1312.6207.pdf)

though this may not be exactly the same problem. In particular, the 9 cylinder
problem allows 3 radii to be selected and then the other 6 are calculated as a
result (meaning the 9 are probably not all identical.) It appears the 7 result
in the initial paper is 7 cylinders of equal radius.

------
abalone
The hippies of Marin County have known this since 1973.

------
ebol4
Don't they just have to not be parallel?

~~~
Retric
No, your thinking 2D lines, but this is 3D space. It's easy to arrange things
so they all miss each other and not have them be parallel. The goal is to have
them touch, but not intersect.

------
pjbrunet
Proof the Internet is a series of tubes.

------
gary4gar

        they built a wooden model to demonstrate their answer — although Bozóki notes 
        that the model doesn’t verify the result because manufacturing errors 
        are much greater than any errors the computer could have made.
    

What's the point, when its not practically possible?

~~~
ColinWright
What's the point of what? Physical models have imperfections and distortions,
so to absolutely know the answer you need the mathematics. Then having done
the mathematics it's satisfying to build the physical model, to see it and to
hold it.

Can you clarify your question? Try to avoid the word "it" because the referent
might not be obvious.

