NavList:

Thanks for the encouragement and generous words, Frank!  Yes friction is the problem, and both Peter Moses and my son suggested 'cars' running on aluminium extrusions with ball bearings so I ordered one to play with. These parts are used to make CNC machines and 3D printers and similar machines. I am now working on the other components. Yes I cannot imagine anyone using it at sea, any more than they would use the construction Peter Moses, Clark Kimberling and I devised for the uncertainty ellipse using ruler and compass! It's more fun and education of course.

Now for your thorny question of least squares. Why not least sum, or some other power? I teach this a lot to a wide range of audiences and my explanations are usually in terms of error distributions, to a smart, not mathematical audience one has to try a few ways depending on how their intuition works. So I don't have one good idea, but one-to-one I would be feeling my way with the student. For some I would illustrate the way different sums of powers behave with outliers?

The highbrow answer is Gauss-Markov theorem: if errors are well-behaved (uncorrelated, constant variance), the Ordinary Least Squares estimator is the Best (lowest variance) Linear Unbiased Estimator (BLUE). But to most people that sounds a bit like "because two important sounding guys say so". You can illustrate this with samples from an error distribution, so a computer simulation with pseudorandom numbers might be convincing.

For my device, I thought the energy minimization might be useful for some people to think about it, and sometimes holding it in ones hands beats a simulation. Maybe its insensitivity near the equilibrium has educational value. Nearby positions are nearly optimal? I might make a simulation too showing realistic springs. Maybe it should have realistic dynamics and springs that go "boing". But this, as you point out, is all about what the least squares solution looks like not why it is a good idea.

Bill