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

[Thomas Klemm]

(01-31-2019 06:07 PM)StephenG1CMZ Wrote:  What is the opposite of a sphere?

The pseudosphere has constant negative Gaussian curvature (rather than the constant positive curvature of the sphere), leading to its name.

Quote:My non-technical guess was all the space outside the sphere (like a torus but in 3D). That has the advantage of being easily visualisable in 3D.

This still has positive curvature.

Quote:But what is the defining characteristic of a sphere that we want the opposite of?
Some answers suggested a negative curvature, and that sounds mathematically interesting, but I'm not yet visualising how that looks.

Locally it looks like a saddle:

Quote:Other answers picked on the number of corners, but I don't think it has a definite number. It seems to me you can make a case for both zero and infinity.

Another answer suggested that a sphere minimises surface area/volume, so its opposite should instead maximise that ratio...But what shape has that characteristic?

Gabriel's Horn has finite volume but infinite surface area:



Cheers
Thomas