## Bounded infinite shapes

For the discussion of math. Duh.

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### Bounded infinite shapes

It is simple, if not quite trivial, to find an example of a two-dimensional shape which has an infinite perimeter binding a finite area, e.g. the Koch Snowflake, or the area bounded by the lines x = 1, y = 0, and y = 1 / x (being half the cross-section of the solid known as Gabriel's Horn). This principle can be extended to three dimensions (and possibly more, but my imagination exceeds my training), producing the same Gabriel's Horn, and Eric Haines' "sphereflake", irrespectively.

Given the existence of such counter-intuitive shapes, I feel I must ask: do there exist shapes in two dimensions which have infinite area bounded by finite perimeters (or the n-dimensional analogues)?
"Maybe there are stupid happy people out there... And life isn't fair, and you won't become happier by being jealous of what you can't have... You can never achieve that degree of ignorance... you cannot unknow what you know." -E. Yudkowsky

Aedl Foxe

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### Re: Bounded infinite shapes

No, because of the isoperimetric inequality.
++\$_
Mo' Money

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### Re: Bounded infinite shapes

Ah! That was simple. Thank you!
"Maybe there are stupid happy people out there... And life isn't fair, and you won't become happier by being jealous of what you can't have... You can never achieve that degree of ignorance... you cannot unknow what you know." -E. Yudkowsky

Aedl Foxe

Posts: 118
Joined: Mon May 02, 2011 6:46 pm UTC

### Re: Bounded infinite shapes

Aedl Foxe wrote:[...] find an example of a two-dimensional shape which has an infinite perimeter binding a finite area, e.g. [...] the area bounded by the lines x = 1, y = 0, and y = 1 / x (being half the cross-section of the solid known as Gabriel's Horn).

I would like to point out that your example has an infinite area (but finite volume if you rotate it around the x-axis).
mfb

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