Thread: Finite volume: Infinite surface area?

While on a lonely walk by the river, a simple question intrigued me. Is it possible to have a closed three dimensional object WITH FINITE VOLUME BUt INFINITE SURFACE AREA?
We may bring the question down to 2D like this:
Is it possible to have a closed figure that has infinite perimeter but encloses a finite area?

yes, take the graph
1/x
then just turn it around the X axis to make it 3d and its done
it got pi volume but infinite area

See Gabriel's horn
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Thanks.
But there appears to be no solution for the problem in 2D. Well, are the probability curves an answer? It seems so, coz whatever limit we take, the area turns out to be one!

Originally Posted by diparnak

Thanks.
But there appears to be no solution for the problem in 2D.

Simply take Zelos's answer and don't "spin" it to create a third dimension.

Originally Posted by Scifor Refugee

Originally Posted by diparnak

Thanks.
But there appears to be no solution for the problem in 2D.

Simply take Zelos's answer and don't "spin" it to create a third dimension.

No, this is infinite. Integrating 1/x from 1 to u is ln(u) - ln(1) = ln(u). As u gets bigger, so does the area.

Originally Posted by diparnak

While on a lonely walk by the river, a simple question intrigued me. Is it possible to have a closed three dimensional object WITH FINITE VOLUME BUt INFINITE SURFACE AREA?
We may bring the question down to 2D like this:
Is it possible to have a closed figure that has infinite perimeter but encloses a finite area?

i have heard that certain fractals have edges of infinite lenght.
it stand to reason if you take such a fractal and spin it about the x or y axis then you will generate a finite volume enclosed by an infinite surface area

Originally Posted by leopold

Originally Posted by diparnak

While on a lonely walk by the river, a simple question intrigued me. Is it possible to have a closed three dimensional object WITH FINITE VOLUME BUt INFINITE SURFACE AREA?
We may bring the question down to 2D like this:
Is it possible to have a closed figure that has infinite perimeter but encloses a finite area?

i have heard that certain fractals have edges of infinite lenght.
it stand to reason if you take such a fractal and spin it about the x or y axis then you will generate a finite volume enclosed by an infinite surface area

Start with an equalateral triangle. For ever edge, attach another equal. triangle half the size of the previous iteration. Keep going. I think this is what you mean?

http://en.wikipedia.org/wiki/Koch_snowflake

Koch snowflake. That's what I was talking about.

Originally Posted by Absane

Start with an equalateral triangle. For ever edge, attach another equal. triangle half the size of the previous iteration. Keep going. I think this is what you mean?

i'm not sure, i read it somewhere that certain fractals have boundries of infinite lenght.

Every closed fractal fits your description. The interesting part is the fact that not only does it have an infinite perimeter, if you look at ANY section of the perimeter, you will find that its length is also infinite.

Originally Posted by leopold

Originally Posted by Absane

Start with an equalateral triangle. For ever edge, attach another equal. triangle half the size of the previous iteration. Keep going. I think this is what you mean?

i'm not sure, i read it somewhere that certain fractals have boundries of infinite lenght.

Yea, like the Koch snoeflake.