# Is there any regular geometrical structure with maximum surface area and minimum volume?

I am interested to know a geometrical structure with maximum surface area and minimum volume. According to me double napped cone may have such property as surface area of a cone is $\pi rl +\pi r^2$ and its volume is $\pi r^2h/3$, where $r$=radius of the base of cone, $l$=lateral height of cone, $h$=height of cone.

• Some fractals suggest themselves – Hagen von Eitzen Apr 19 '13 at 5:20

I know one of such structures. It is a plane. :-)

• Yes I m agree with u, but sorry for that I forgot to mention the dimension of the above geometrical structure and actually it should be at least 3-dimensional. – Lakshman Mahto Apr 20 '13 at 10:21
• It seems that this restriction does not change the infimum of volume (0) and the supremum of surface area $(\infty)$, but makes them unattainable by bounded convex bodies. But the volume can be done abitrary large, and, simultaneously, the surface area can be done arbitrary small nonzero, by any sufficiently “flat” and “stetched” body. For instance, by the pyramid with the vertices $(-n,-n,0)$, $(-n,n,0)$, $(n,-n,0)$, $(n,-n,0)$ and $(0,0,1/n)$. – Alex Ravsky Apr 20 '13 at 18:09
• Thank u for such a beautiful example. As n approaches to infinity volume of the pyramid tends to zero and the surface area approaches to infinity. – Lakshman Mahto Apr 20 '13 at 19:06
• Oops, for the volume zero limit the fifth vertex should be $(0,0,1/(n^3))$ instead of $(0,0,1/n)$. – Alex Ravsky Apr 20 '13 at 19:10
• Yes actually volume of pyramid is a^2*h/3, if fifth vertex is (0,0,1/n^3) then volume becomes 2/3n. – Lakshman Mahto Apr 21 '13 at 9:27

look up a Menger Sponge on wikipedia

• Hi, welcome to math.SE. Usually when writing answers here, it's best to add a bit of explanation beyond just "go look here." Even if it's only this much you can also improve it by furnishing the link you are speaking of. You can do this by wrapping the linked text in square brackets and following them immediately with the url wrapped in parentheses. – rschwieb Jun 17 '13 at 13:47
• Please try to describe as much here as possible in order to make the answer self-contained. As it stands, this is more of a comment than an answer. – robjohn Jun 17 '13 at 14:34

You might want to use a labyrinth (taking a 2D one and giving its walls thickness and height to make it 3D). the more complex it is, the more surface you have, while keeping the volume under control (by limiting the walls' thickness and height).

But the ideal would be a fractal : their surface tends to the infinite while their volume tends to zero. these are objects that are infinitely fractioned (they have a "broken" aspect in all scales) examples: Sierpinsky pyramid, menger sponge as user82834 mentioned, fractal trees, Mandelbulb (I'm not sure about the last one though) and many others.

if you want a "regular" geometrical figure, you can limit a fractal to its first iterations, so you won't have any infinities appearing in your calculations.