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Thread: Finite volume: Infinite surface area?

  1. #1 Finite volume: Infinite surface area? 
    Forum Freshman diparnak's Avatar
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    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?


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  3. #2  
    Forum Isotope Zelos's Avatar
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    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


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  4. #3  
    Forum Radioactive Isotope MagiMaster's Avatar
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    See Gabriel's horn
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  5. #4  
    Forum Freshman diparnak's Avatar
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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!
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  6. #5  
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    Quote 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.
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  7. #6  
    Forum Sophomore Absane's Avatar
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    Quote Originally Posted by Scifor Refugee
    Quote 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.
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  8. #7 Re: Finite volume: Infinite surface area? 
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    Quote 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
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  9. #8 Re: Finite volume: Infinite surface area? 
    Forum Sophomore Absane's Avatar
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    Quote Originally Posted by leopold
    Quote 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?
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  10. #9  
    Forum Sophomore Absane's Avatar
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    http://en.wikipedia.org/wiki/Koch_snowflake

    Koch snowflake. That's what I was talking about.
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  11. #10 Re: Finite volume: Infinite surface area? 
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    Quote 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.
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  12. #11  
    Forum Junior Vroomfondel's Avatar
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    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.
    I demand that my name may or may not be vroomfondel!
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  13. #12 Re: Finite volume: Infinite surface area? 
    Forum Sophomore Absane's Avatar
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    Quote Originally Posted by leopold
    Quote 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.
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