
Gabriel's Horn - wz1000
https://en.wikipedia.org/wiki/Gabriel%27s_Horn
======
dubin
Also check out the Menger sponge, which has infinite surface area and zero
volume:
[http://en.m.wikipedia.org/wiki/Menger_sponge](http://en.m.wikipedia.org/wiki/Menger_sponge)

~~~
relicscattergun
I think Sierpinski's triangle may also be of interest here, which, contrastly,
approaches zero surface area:
[http://en.wikipedia.org/wiki/Sierpinski_triangle](http://en.wikipedia.org/wiki/Sierpinski_triangle)

~~~
gohrt
not constrast; it's analogous, in lower dimension.

Sponge has infinite area and 0 volume. Sierpinski's triangle has infinite
perimiter and 0 area.

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jveld
I've thought it was paradoxical that infinitely long curves could have finite
integrals ever since I first took calculus. For example, the integral of 1/x
is |ln x|.

I wonder why it takes three dimensions before people start getting upset.

~~~
parallel
Or even just a convergent infinite series. Infinitely many positive numbers
that sum to a finite number.

~~~
thaumasiotes
People don't tend to find that unintuitive. The Greeks had a nice geometric
example in a square of side length 1:

    
    
        ┌┬┬─┬───┬───────┐
        ├┘│3│   │       │
        ├─┘-│   │       │
        │ 64│   │       │
        ├───┘   │       │
        │       │       │
        │ 3/16  │       │
        │       │       │
        ├───────┘       │
        │               │
        │               │
        │               │
        │       3/4     │
        │               │
        │               │
        │               │
        └───────────────┘
    

3/4 + 3/16 + 3/64 + 3/256 + ... is easy to visualize as successively filling
in three quarters of an ever-smaller residual square. Intuitively, no matter
how finely you detail it, you're never going to stop fitting inside the
original area-1 square.

edit: better text art

~~~
lmm
Now squeeze the top-left piece so it's half as wide but twice as tall (i.e.
the same height as the full rectangle), and do this recursively. Same area,
right? Then stack the L shapes on top of each other. Then you have one of
these horns.

~~~
thaumasiotes
Huh? The horn is a three-dimensional object; the square exists in 2-space. You
can't make the horn from pieces of the square even if you allow deformation.

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mgob
I always describe this to "non-mathy" people when they ask what could possibly
be fascinating/beautiful/etc about math. I'd like to think I've changed at
least a mind or two.

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TeMPOraL
I wonder what sound it would make. Can we model such a horn assuming _a_ from,
say, 1cm to some value _p_ and simulate it?

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currentoor
Depends on how many of the laws of physics you want to ignore. The horn has
infinite length and sound waves have a finite speed (assuming sound is defined
as the usual compression wave through a medium). So a sound wave starting at
the narrow end would take an infinite amount of time to reach the other end.
But the sound it self shouldn't be anything special.

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davidrusu
This reminds me of a Putnam problem from a few years back, was something along
the lines of:

Construct a set of discs in R^2 s.t. no infinite straight line can be drawn
without intersecting at least one disc, and the sum of the areas of all the
discs is finite.

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cousin_it
Take a set of discs with radii 1/n. Their total area is finite but the sum of
radii is infinite, so we can just use them to cover the X and Y axes. Yeah,
this is pretty similar to Gabriel's horn.

