(no subject)

[simont]

Oh yes, and I've just added some more Stuff to my website. It's now getting to the point where I might have to fiddle with the front page so that it has a usable index; all this chatty text introducing each of my subpages is fine if you're reading it because you want to know about me, but even I'm increasingly finding it a pain when I actually want to find a specific page.

Still. For anyone with any interest in polyhedra, or even simply with a desire to look at some pretty pictures or see what weird mathematical things I've been up to recently when I should have been lying on my sofa watching DVDs, you can now go and look at http://www.chiark.greenend.org.uk/~sgtatham/polyhedra/.

Now, with any luck, I ought to be able to actually relax for a bit. These momentary obsessions tend to take me over totally for a week or so, but then leave me in peace after that…

Comments

[simont]

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.

[ptc24.livejournal.com]

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.