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    objectundefined's Avatar
    objectundefined Posts: 45, Reputation: 2
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    #1

    Apr 21, 2010, 05:13 PM
    What is the theoretical 3d shape that is all surface area?
    [br]What is the theoretical 3d shape that is all surface area?[/br]
    InfoJunkie4Life's Avatar
    InfoJunkie4Life Posts: 1,409, Reputation: 81
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    #2

    Apr 22, 2010, 07:52 AM

    I would imagine it would be a 3d fractal. Fractals are said to have an infinite perimeter. Turning it into a 3d object would be likely to have an infinite surface area.

    Mandelbulb

    Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal
    Clough's Avatar
    Clough Posts: 26,677, Reputation: 1649
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    #3

    Jul 1, 2010, 11:49 PM
    Wow! Amazing images on that website, InfoJunkie4Life!

    Thanks!
    objectundefined's Avatar
    objectundefined Posts: 45, Reputation: 2
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    #4

    Jul 2, 2010, 12:27 AM
    Cool images, but not what I'm looking for. It's always presented as a sort of jar. A vase-shaped, glassy object that had no volume and was all surface area.
    galactus's Avatar
    galactus Posts: 2,271, Reputation: 282
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    #5

    Jul 2, 2010, 02:22 AM

    I think what you are referring to is called Gabriel's Horn.

    If we take the unbounded region lying between the x-axis and , we get a solid with finite volume but infinite surface area.

    The volume is given by:



    The surface area is given by

    .

    This is an improper integral that diverges and thus shows infinite surface area.

    Because on the interval ,

    and the improper integral diverges,

    we can conclude that the improper integral also diverges. Therefore, the surface area is infinite.
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    elscarta's Avatar
    elscarta Posts: 118, Reputation: 20
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    #6

    Jul 2, 2010, 09:08 AM
    While this may not be the jar you are looking for the Menger Sponge is also a 3D object with zero volume, making it all surface area.
    dmcdrmtt's Avatar
    dmcdrmtt Posts: 1, Reputation: 1
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    #7

    Aug 15, 2011, 04:15 PM
    You're looking for a Klein bottle - a 3D object with no interior. http://www.google.com/search?q=klein...2&ved=0CEEQsAQ

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