[br]What is the theoretical 3d shape that is all surface area?[/br]
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[br]What is the theoretical 3d shape that is all surface area?[/br]
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
Wow! Amazing images on that website, InfoJunkie4Life!
Thanks!
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.
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.
Becauseon the interval
,
and the improper integraldiverges,
we can conclude that the improper integralalso diverges. Therefore, the surface area is infinite.
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.
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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