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-   -   What is the theoretical 3d shape that is all surface area? (https://www.askmehelpdesk.com/showthread.php?t=466232)

  • Apr 21, 2010, 05:13 PM
    objectundefined
    What is the theoretical 3d shape that is all surface area?
    [br]What is the theoretical 3d shape that is all surface area?[/br]
  • Apr 22, 2010, 07:52 AM
    InfoJunkie4Life

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

    Thanks!
  • Jul 2, 2010, 12:27 AM
    objectundefined
    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.
  • Jul 2, 2010, 02:22 AM
    galactus
    1 Attachment(s)

    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.
  • Jul 2, 2010, 09:08 AM
    elscarta
    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.
  • Aug 15, 2011, 04:15 PM
    dmcdrmtt
    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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