Geometry:what is the opposite of a sphere?& what shape maximises surface area/volume?

[berndpr]

Hello!

You can use the Koch curve (see https://en.wikipedia.org/wiki/Koch_snowflake) for finite area but with infinite perimeter in a 2d plane, if you start with a triangle.

I think you can expand this principe to 3d objects like the tetrahedron.
Each of the triangle faces can split in 4 similar but smaller triangle (https://upload.wikimedia.org/wikipedia/c...at.svg.png).

If you replace for the inner triangle a new smaller tetrahedron again, you get the a 3d version of the Koch curve.

And if we repeat this process infinite we get a solid with finite volume, but infinite surface.

I think you can do this with nearly spheric object like the football (https://en.wikipedia.org/wiki/Buckminsterfullerene) too. You must only divide the faces in similar pieces and replace it with a part of a smaller version of a 3d object. The only condition is: The smaller 3d object must have plane faces like the original object.

Don't hesitate to correct me, if I'm not right.

Bye
Bernd