Nice experiments.
I confess I’ve never actually tried this with measurements physically myself, so it’s nice to see confirmation.
I imagine getting much closer to the theoretical limit would require strict laboratory conditions, as it’s the equivalent in my green sketch above of trying to balance the ball on a vanishingly small bump on the side of a steep hill, so little vibrations would tip it over.
Area minimization with a volume constraint (as with LiveSoap with ‘UseVolume’ on) should produce CMC surfaces. Minimizing edge lengths alone indeed won’t give CMC surfaces.
Some shapes are a bit easier to achieve as stable CMC surfaces - for example, with minimal surfaces most shapes with handles or tunnels collapse, but the additional volume constraint can make some of these stable.
However, lots of the spatially interesting CMC surfaces such as the ones in that Polthier paper you link are unfortunately not physically stable (despite being true CMC solutions). These critical point solutions are a bit like geodesics between 2 points on a sphere. Generally there will be 2 different arcs between them on the same great circle, both these arcs are geodesic, but an elastic cable would only be stable on the shorter one.
These unstable solutions aren’t as easy to model with Kangaroo - it will generally be like physical modelling, so ones that pop there will pop in simulation.
Finding CMC surfaces within fixed boundaries is relatively easy, but anything closed tends to just pop to a sphere, and trying to find something periodic with handles, the handles quickly pop.
Sometimes it is possible though. For instance, if you know some symmetry properties of the correct solution, enforcing these can sometimes be enough to stabilise it without distorting it from CMC.
I am interested in finding better ways to simulate the unstable minimal and CMC solutions though. For discrete quad versions you can use the Koebe polyhedron method I was talking about recently over in this thread.
For something more general it might require some changes to the solver though.