What is the theoretical 3d shape that is all surface area?

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 (http://www.skytopia.com/project/fractal/mandelbulb.html)

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 y=\frac{1}{x}, we get a solid with finite volume but infinite surface area.

The volume is given by:

V={\pi}\int_{1}^{\infty}(\frac{1}{x})^{2}dx={\pi}

The surface area is given by

S=2{\pi}\int_{1}^{\infty}\frac{1}{x}\sqrt{1+\frac{ 1}{x^{4}}}dx.

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

Because \sqrt{1+\frac{1}{x^{4}}}>1 on the interval [1,{\infty}),

and the improper integral \int_{1}^{\infty}\frac{1}{x}dx diverges,

we can conclude that the improper integral \int_{1}^{\infty}\frac{1}{x}\sqrt{1+\frac{1}{x^{4} }}dx also diverges. Therefore, the surface area is infinite.

While this may not be the jar you are looking for the Menger Sponge (http://en.wikipedia.org/wiki/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+bottle&hl=en&lr=&client=firefox-a&rls=org.mozilla:en-US:official&biw=1280&bih=691&prmd=ivns&tbm=isch&tbo=u&source=univ&sa=X&ei=cKdJTqjCEtSlsQKkpK2vCA&sqi=2&ved=0CEEQsAQ