This all reminds me about some inorganic chemistry lectures I once went to. IIRC small clusters of metal atoms like to form constrained-to-sphere shapes, with a strong preference for lots of triangles.

I wonder about the other good way of arranging 8 points: an octahedron with two opposing faces being 'capped' by an extra vertex. Does your algorithm always get the global minimum?

I have no particular reason to think that my algorithm will always get the global minimum - it just starts with a random point set and runs a single naive convergence from there. However, I've never yet seen it give different answers on successive attempts. (It begins with the points uniformly and randomly distributed over the sphere, so if there were to be multiple significantly different solutions with catchment areas of roughly equal sizes, I'd expect to have seen one by now.)

"an octahedron with two opposing faces being 'capped' by an extra vertex"

In other words, a cube! :-)

At least, the only real difference between what you describe and a cube is that each face is potentially folded into two triangles, one part of a 'cap' and one an original face of the octahedron. I suspect that for symmetry's sake you'd want to fold the faces by little or nothing rather than by lots, otherwise you end up bringing points closer to each other than they need to be; but if you start with a cube, it appears much more likely to buckle into the shape I show, because that's quite a long way from the cube and presumably has a significant improvement in energy level.

Now you mention it, it is a fairly trivial distortion of a cube. Trouble is, I can't remember what the circumstances were when that shape was favoured over the square antiprism, so I can't say more.

you look just like somebody i met at the selwyn open day in 1996, i *think* - were you there
? *curious*

I had short hair back then. But yes, I was there at an open day in 1996. The world is small.

it is good to see people *growing* hair instead of chopping off, as seems to be a frequent occurence in the early twenties.. *sigh*

that open day was one of my most depressing experiences of 1996. not quite sure why, but i was very happy to apply ahywhere except!

*clutches protectively at slowly-growing hair*

wish more peopoe would clutch at theirs!