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I was just reading about Gabriel Horn and just wanted to know if there were others. Meaning with infinity surface area but finite volume, or infinit volume but finite surface area.

I have tried to find a cross section of the Klien bottle, but the equations I have found for it are above my level of math. Studying for the AP BC Exam.

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Depends on what you mean by "like", but have you checked the "See also" section a Wikipedia? – Hagen von Eitzen Dec 10 '12 at 14:58
where either the surface area or volume is finite, but the volume or surface area is infinite. – yiyi Dec 11 '12 at 3:01
The idea of volume and surface area of a Klein bottle probably don't have much to do with this particular question, as the Klein bottle can only live in 4 dimensions. – Gyu Eun Lee Dec 11 '12 at 3:07
So these Kline bottle's are fakes? – yiyi Dec 11 '12 at 3:13
They aren't actual Klein bottles. They are the best we can do in three dimensions, but restricted to three dimensions the Klein bottle is forced to intersect itself, something surfaces are not allowed to do. – Gyu Eun Lee Dec 11 '12 at 3:16
up vote 3 down vote accepted

Well, first one has the following fact:

If one makes the corresponding "Gabriel's Horn" by rotating the region under $\frac{1}{x^p}$ from $x=1$ to $x=\infty$ around the $x$-axis, then:

  1. The volume is finite iff $p>\frac{1}{2}$

  2. The surface area is finite iff $p > 1$ so in particular

  3. The Horn has finite volume and infinite surface area for any $p$ between $\frac{1}{2}$ and $1$, where $p$ cannot be $\frac{1}{2}$ but it can be $1$.

The proof is simply several uses of the comparison theorem.

One cool thing that pops out is that the standard Gabriel's horn $(p=1)$ actually gives the minimal volume Gabriel's horn which still has infinite surface area. If you shrink the volume at all (i.e., $p$ goes up), then the surface area becomes finite.

There are, of course, variations on this idea. For example, one can add some small bumps to the graph of $\frac{1}{x}$. Note that bumps will tend to increase the surface area (so it will stay infinite), but if the bumps are not too frequent or too high, then the volume will still converge. (Concretely, imagine rotation $\frac{\sin x}{x}$ with $1\leq x < \infty$ around the $x$ axis).

Finally, there are some other shapes which are in no way related to Gabriel's horn. For example, the Menger Sponge has finite volume but infinite surface area. You start with a cube of volume $1$ (so, very finite), and start cutting pieces out. Of course, this only lowers the volume, so the volume is finite at the end of the day. On the other hand, cutting pieces out increases the surface, and at the end up the day it's infinite.

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The Klein bottle stuff is interesting but also irrelevant; I'll get around to the question itself now.

Yes, there do exist other surfaces that share this property with Gabriel's horn. Here's one called Gabriel's wedding cake: (Gabriel's got a thing for finite volumes and infinite surface areas it seems. Makes sense for the horn, would be mighty hard to blow a horn of infinite volume - acoustics and all - but I'd have elected for a cake of infinite volume over one of infinite surface area myself.)

There's also the Koch snowflake - a curve of infinite length that bounds a planar region of finite area.

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