Rich mentioned the Science News piece about unistable objects. The write-up (in the Math Trek column, which Julie Rehmeyer just took over when Ivars Peterson moved to the MAA) is based on the articles by Várkonyi and Domokos in the most recent Intelligencer, where it's clear that the point is that they have found the first "uni-unistable" objects, ones with only one stable equilibrium *and* only one unstable equilibrium. That's why the Conway-Guy unistable 19-hedron doesn't count, of course: it has three unstable equilibria. --Michael On 4/10/07, Richard Schroeppel <rcs@cs.arizona.edu> wrote:
Science News' Math Trek has an interesting article this week, by Julie J. Rehmeyer. It's about a unistable object discovered by Gabor Domokos and Peter Varkonyi.
http://sciencenews.org/articles/20070407/mathtrek.asp The SciNews page has a picture, and pointers to three of their tech reports.
A unistable object is a 3D shape with only one stable equilibrium point. It's easy to do this if you allow the object to contain an osmium mascon under just under the surface of a curved face, or equivalently if you allow holes (or voids). The problem is interesting if you require a convex object of uniform density. We've discussed it on this list a couple of times.
My particular inspiration may go back to the early space program: the astronauts would come back in a small space capsule, parachuting into the ocean. The capsule designers had to plan for two possible orientations of the capsule while it was bobbing in the ocean, waiting for recovery.
The new object has curvy faces with various edges, reminiscent of a "surface of constant width". It has one stable and one unstable balance point, although the picture looks like some of the edges might also contribute more unstable balance points. It vaguely resembles a turtle, which has lead the authors to start playing with turtles.
There is a prize offered for a polyhedral solution, but it's $10K/face-count. The authors don't expect to pay much. There's no discussion of the obvious first move, simply taking their rounded solution and using sandpaper to make a lot of tiny flat faces.
(unrelated) news item from last week: Fermilab has been building magnets for CERN's Large Hadron Collider, and one failed a pressure test badly, "blowing up" in the tunnel. No injuries, damage assessment in progress, Fermilab appropriately embarrassed, lots of copies of the magnet will need rework. First light was expected this fall, noone's admitting to a schedule slip yet.
Rich rcs@cs.arizona.edu
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