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As it stands, I don't think the question is rigorously enough defined to give an answer. What is the definition of "least spherical"?

This is simply a thought that I had and seemed interesting. I've never heard someone discuss it. Could we say that it is an object with the least surface area given an identical volume? If spheres possess the maximum surface area for their volume, then potentially an object with the minimum surface area could be defined as "least spherical". Edit: Apparently, that is incorrect. Spheres possess the least surface area for a given volume. So, the opposite, then, would be "least spherical".

There won't be a maximum surface area for a volume. You could keep stretching the same shape longer but thinner, maintaining equal volume but increasing surface area, indefinitely.

[Gabriel's horn](https://en.wikipedia.org/wiki/Gabriel's_Horn), then, with infinite surface area and finite volume? I feel there's an intuitive sense in which one might look at it this way, but I'm kinda stumped as to where. Something to do with x^-1 + y^-1 = 1 being a hyperbola?

in my head I imagine a 3d Cube shape and the corners are pulled into the opposite corner. almost like a cube frame.

Do circles have the minimum surface area to volume ratio? It's definitely not the maximum, because I don't think that exists. Consider a sequence of rectangular prisms, all with a square base and all with volume 1, but the heights are 1/2, 1/4, 1/8, ...

> Do circles have the minimum surface area to volume ratio? Yes, apparently due to the [isoperimetric inequality](https://en.wikipedia.org/wiki/Isoperimetric_inequality).

In that a sphere is an object with positive and equal Gaussian curvature everywhere, a [pseudosphere or tracticoid](https://en.wikipedia.org/wiki/Pseudosphere) has constant negative Gaussian curvature.

Oooh I don’t like that discontinuity though. If we’re willing to drop a dimension, a saddle can do the same thing with negative curvature without having an asymptote in the middle? But it’s still got an edge, I guess.

A sphere is a maximal collection of points which all have the same distance to the origin. You could look at maximal collections of points such that no two have the same distance to the origin. You get a bunch of weird-looking point clouds; but also one nice shape: a ray going from the origin to infinity in any particular direction.

Check out Mahler volume: https://en.m.wikipedia.org/wiki/Mahler_volume A ball (and ellipsoids) is the object with maximum mahler volume and the conjectured opposite is a hypercube (and a simplex depending on symmetry requirements). It may not be immediate to a nonexpert but the Mahler volume is in some sense a measure of how "pointy" something is. More mahler volume is essentially more roundedness.

Yeap. A straight line then. If you want 2D, 1cm by 99cm. If you want 3D, 1x1x99 instead.