
Thought Experiments in Mathematics: Gabriel's Horn - mgdo
http://fermatslibrary.com/s/gabriels-horn
======
kps
The link is an image of a print-formatted copy of the Wikipedia article
[https://en.wikipedia.org/wiki/Gabriel%27s_Horn](https://en.wikipedia.org/wiki/Gabriel%27s_Horn)
which is more easily read directly on many devices.

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Jun8
This answer on Math SE illustrates the concept of finite volume-infinite area
quite nicely
([http://math.stackexchange.com/a/14632/14643](http://math.stackexchange.com/a/14632/14643)).

Note that the converse is proven only for surfaces of revolution and for
differentiable functions. Finding a pathological counter example that violates
these assumptions would be interesting.

~~~
Chinjut
The converse is impossible in any context where the isoperimetric inequality
is applicable.

~~~
tgb
Good point. Though the Wikipedia article on it gives the most general form as
requiring our region to have closure with finite Lebesgue measure. This is
necessary of course by the example another corner brought up about swapping
the inside and outside. But that means it exactly can't rule out the existence
of regions with infinite volume (outside and inside) and finite surface area!

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roywiggins
I'm a big fan of the Alexander horned sphere, which is even weirder. The
interior is simply connected (like the interior of a sphere) but the exterior
isn't.

[https://en.wikipedia.org/wiki/Alexander_horned_sphere](https://en.wikipedia.org/wiki/Alexander_horned_sphere)

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Chinjut
For what it's worth, this is a 3d version (finite volume, infinite surface
are) of a phenomenon which might be more easily grasped in 2d (finite area,
infinite perimeter).

And this idea of finite area described by y = f(x) [between this curve and the
x-axis, say], with infinite perimeter, amounts to just the same thing some
function f whose integral over an infinite range (e.g., from x = 1 to
infinity) is finite. If you prefer to think discretely, this is essentially
the same phenomenon as an infinite series with finite sum (splitting our curve
into the block from x = 1 to 2, the block from x = 2 to 3, the block from x =
3 to 4, etc., the total area is the sum of the series of areas in these
blocks, while the total perimeter is automatically infinite as at least length
1 on top and bottom is contributed in each of these blocks). So, consider, for
example, y = 1/x^2, from x = 1 to infinity: infinite perimeter but finite area
(as its antiderivative is -1/x + C, which only increases by a finite amount
over this range; in the same way, differentiating any function with a finite
asymptote yields examples, and every example comes in essentially this
fashion).

In moving to 3d and making a surface of revolution out of our starting curve,
we are now integrating πf^2 instead of f itself, so we need f^2 to yield a
convergent integral rather than f itself, but otherwise everything is just the
same. Thus, we can get away with f(x) = 1/x, as in Gabriel's Horn, but also,
just as well, with f(x) = x^{-p} for any p > 1/2.

Not that these are the only such things; again, any convergent infinite
integral or series yields examples. So, just as well, y = 1/2^x or such things
would work.

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sixstringtheory
I remember finding this gem in my calculus book's exercises, and toiled over
it and debated with my professor and lost sleep over it. Well, I didn't
disprove it, but I learned a lot of good stuff along the way :)

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zeroer
The most interesting thing about it is that at one point people thought it was
a paradox.

~~~
Someone
It took lots of very intelligent mathematicians to realize that infinities are
weird.

Even today, it takes quite some schooling to come to terms with some of the
_easy_ examples.

For example, I expect that easily over 95% of university graduates disagree
with the statement _" There are as many rational numbers as there are prime
numbers."_.

I even fear that is true for the easier to believe _" 0.9999999... = 1"_

~~~
llamaz
There probably is _some_ mathematical formalism out there where there are less
prime numbers than natural numbers.

For example, most mathematicians would agree that there are as many reals in
[0, 1] as there are in the real line. But according to a different definition
of size, namely the most famous measure from measure theory, the Lebesgue
measure --- the length of [0, 1] is 1 while the real line is undefined (I
think). The person on the street would probably prefer the Lebesgue measure as
more intuitive.

I guess my ultimate point is that imprecise statements are harder to prove
wrong. There's a quote, I don't remember who said it, that you should mistrust
precise statements rather than imprecise ones, because it's precisely precise
statements that can be proven wrong.

edit: found the quote:
[http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329....](http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329.html)

~~~
Someone
_" There probably is _some_ mathematical formalism out there where there are
less prime numbers than natural numbers."_

You could call Z[n] ?for any composite n? that (in Z[4], the multiplication
table only contains 0, 1, and 2, so 3 is prime there; Z[p] for prime p gives
you p different numbers and zero primes) but I think those are the only ones.
If you accept that infinity exists you get Hilbert's hotel, which gets you all
those paradoxes, which after lots of sleepless nights leads to the only
logical conclusion that giving up intuition about infinities is the best way
out.

If you don't accept that infinities exist, there must be a largest integer M,
and you get to decide what M+1 or 2M are. That leads either to Z[n], to K&R's
_undefined behavior_ , which is so ugly no mathematician would dare publish it
:-), or to some formalized variant of it that isn't Z[n].

I'm not sure I would call the values of Z[n] _natural_ numbers, though, as
that feels like it requires having negative numbers, too. Hm, maybe a shifted
Z[n] would work. If you replace {0,1,2,3} by {0,1,2,-2,-1} in Z[5], you have
two negative and three natural numbers in your universe, none of which is
prime.

I doubt that any of this kind of mathematical hair-splitting would bring
aboard those who have trouble with grasping _0.999999... = 1_ , though :-)

~~~
SomeStupidPoint
Can you not make a "tropical Zn" with elements 0 - (M-1), and a "big" element,
representing anything M or larger, including infinite values?

I dont know that it would be exactly analogous to normal Z[M+1], and might
have useful properties for some kind of geometric or combinatorical modeling.
(My hunch is any time you want to be capable of carrying a "crossed threshold"
flag as well as a value, and have that cascade through calculation.)

It would also model systems where you can get to infinity in finite steps, but
can't traverse back. Not sure if those are useful in the abstract, though.

~~~
Someone
IEEE 754
([https://en.wikipedia.org/wiki/IEEE_floating_point](https://en.wikipedia.org/wiki/IEEE_floating_point))
does that (for floating point, but the integer variant would be easily
defined), with a few extensions. You need "small", too, for example, and
"don't have the faintest idea" for when somebody subtracts "big" from "big".

That's pragmatic, highly useful, but a nightmare for mathematicians. You lose
invariants such as _x+y-y=x_ (associativity and communicativity, in general),
so you're no longer talking of a group (I don't know of research on 'almost
groups')

Compiler writers may happily make matters worse by assuming those laws still
hold, with the effect that the same computation may overflow or not on
different CPUs, under different compilers, compiler settings, or even the same
compiler in the same compilation run.

~~~
llamaz
"almost groups" are called semigroups. Somehow people have written multiple
books on the subjects.

~~~
Someone
Oops and thanks. I don't know how I managed to not fish that out of my memory
while thinking whether half groups or incomplete groups existed. I even
thought of the term quasigroup (or did I think of 'almost group'?) but
unfortunately didn't google it, as it would have learnt me they exist.

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keithnz
For programmers it is helpful to visualize the horn such that the volume =
features and the surface = possible bugs

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kevinwang
Is this similar to the Dirac Delta function, which bounds a finite area, with
infinite length? Could one say they are different dimensional analogues of
each other?

~~~
thomasahle
The Dirac Delta isn't really well defined from a calculus point of view. It's
basically a half line in space, so its area should be 0.

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mkagenius
How is this different from this: "a plain sheet of paper can have infinite
surface area and finite (0) volume" ?

~~~
danbruc
An ideal sheet of paper, a plain, is a two dimensional object and has by
definition no volume. In case of Gabriel's horn we really have a three
dimensional object bounded by a two dimensional surface. The bounding surface
itself has no volume just like a plain.

~~~
ezequiel-garzon
I cannot resist: it's a plane, not a plain.

~~~
danbruc
Damn it, I even looked it up because I was unsure. But the German word Ebene
translates to plane and plain and I missed that.

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blippityBlop
Okay, tra la la. But can we all take note that the volume is PI, and that PI's
decimal places are also infinite.

In this moment, I take issue with the idea that PI represents a "finite"
volume.

PI, is more accurately a "known-volume-other-than-infinity."

Meanwhile, angels dance upon the head of a pin somewhere.

~~~
drdeca
You can just re-scale it to have volume 1 if you want.

Also, uh, you realize that pi is less than 5, and more than 0, right?

If you draw a circle with radius 1, the area would be pi just as much as the
volume of this at some scaling.

Would you say that such a circle has infinite area? That seems like it would
be refusing to use a term as it is generally used. You can object by saying
that circles do not exist, I suppose. That would be consistent.

But saying that a unit circle has infinite area is either false, or using a
term in a nonstandard way.

Similarly, you could reject such a horn (perhaps because it isn't bounded),
but calling the volume infinite is like calling the area of a unit circle
infinite.

Maybe you just object to abstract objects in general?

If so, alright. People are still going to study them though.

~~~
blippityBlop
So, if I was going to argue about things not existing, the concept of pi, the
irrational number, would be my first target. It's just an artifact of bean
counting that people chase their tails over.

To say that pi is a non-repeating fraction with infinite decimal places, is to
say that we cannot ever precisely know the boundary and definition of a
circle, forever zooming in on a point never met.

That we find circles, and capitulate to what must be their true or approximate
location is a cheat, and mere luck at the integrity of atoms or planck space
or what have you.

But still, it's garbage in the same way that the divide-by-zero rule is
garbage. Granted, dividing something into nothing isn't a practical operation.
But, I can divide zero into as many parts as I like, and it's still zero.

So, if people want to keep stacking infinitesimally small tenths, hundreths,
thounsandths, millionths and so on, to the end of any particular number,
forever, well, forever sounds like an infinite quantity to me, and I'll argue
to point that out.

Unroll the circumference of a circle. Where does it end? Approximately near
_X_? Approximately???

Tell me the area of a circle. Approximately within this fuzzy boundary?
Approximately???

I am dubious.

~~~
drdeca
Are you objecting that infinite precision or irrational numbers do not
physically exist?

That may be. Or it might not. Whether (physical) space is infinitely divisible
or not is, I think, an open question. I think it is even plausible that the
answer might be that it can be described equally well both ways? Not sure.

I don't mind thinking about abstract objects which do not physically exist.
(Though, I generally consider abstract objects to exist in some sense, though
you may disagree, and that's fine.)

If you don't care to think about them, that's fine also.

I believe that thinking about them is often useful. Treating pi as existing
allows for reasoning about things more efficiently, even if it turns out that
it results in inaccuracies in predictions about the physical world which are
far smaller than our ability to measure/notice. (If we can't notice the
results of the error, it doesn't seem to be much of a problem.)

That doesn't mean you have to accept pi as actually existing. I think it would
be to your benefit to at least accept its use though. I do consider it to
actually exist.

