
The Clifford torus is obtained by rotation of a circle of radius R_{1} around an axis placed at a distance R_{2}=2^{1/2} R_{1} apart. Its reduced volume is 0.71 and minimizes the curvature elastic energy among the surfaces with topological genus 1.
This is a result still to be proven mathematically. Our observations are just a kind of experimental tool to solve such problems.


The Dupin cyclides are obtained from the Clifford torus by sphere inversions. As a result, they also minimize the curvature elastic energy.
Well, I mean this is not trivial, but this curvature energy is conformally invariant, and sphere inversions are one kind of conformal transformations.
This peculiar one has a reduced volume of roughly 0.8.


Another Dupin cyclide, with a reduced volume of about 0.9. To be precise, there is a whole (continuous) family of surfaces minimizing the curvature elastic energy between the Clifford torus (v_{red} = 0.71) and the infinitesimally pearced sphere (v_{red} = 1).
So, next time you buy a sphere, check its topological genus.


Theoreticians are more useful than one might believe at first sight. For instance they are very brilliant people who are no competitors to experimentalists...
But sometimes they miss some of the subtleties of Miss Nature, which we simply have to look at. This shape is an example of that, calculated starting from the experimental observation. This stomatoid torus is very stable, indeed.

 This shape deserves the same commentary.


After genus 1 (tori), comes genus 2 (bitori ?). Anyway, if one creates two holes and minimize the curvature elastic energy, one gets something like this button.
It was first calculated by Rob Kusner and his friends with the Surface Evolver program, and first shown to me by Eric Boix, then Ph D student at the Ecole Polytechnique (Eric, where are you now ?).
I couldn't believe it ! We had observed this kind of shapes for months if not years, and couldn't imagine they could be easily computed. There was some little things to do to come to actual membranes, but at least the basis was here.


The Baron de Dupin couldn't dream about things like this, but he ought to be cited here, since this surface is nothing else than the equivalent of a Dupin cyclide for the previous button. This button shape is in fact associated with a whole 2parameters family of shapes with the same minimal energy.
This surface is called the Lawson surface, from the name of a mathematician who studied it in S^{3}, the unit sphere of the 4D space. It has a threefold axis, and only two holes, that is two handles, despite the appearance. Just think a little bit...
