Last week I was privileged to attend a small puzzlement including Rob Pike (organizer), Wei-Hwa Huang, David Hagelbarger, Scott Schwartz, and John Lamping. David had machined an aluminum biroller. (He also made an aluminum version of my Twubblesome Twelve.) Wei-Hwa pointed out an obvious derivation of the volume: if the biroller with its inscribed sphere rests on a pole, all the horizontal cross-sections are squares with inscribed circles. So volumes(biroller/sphere) = areas(square/circle) = 4/pi. Archimedes may have seen this and not (completely) invented integral calculus after all. The same trick applies to moments-of-inertia(biroller/sphere) = 16/3pi. John was able to spin the biroller on an "equatorial corner", so that it swept out the union of a bicone and prolate spheroid. My insight- free double integral says this axis has exactly half the moment of inertia of the polar one, 64r^5/45. The opaque region of the corner-spinning biroller resembles a prolate spheroid, but is actually the intersecion of the spheroid and the bicone. Is there a name for these "inscribed solids of rotation"? I ran a long demo on my battery-powered laptop, using a conference room projector instead of the laptop screen. At the end I was amazed to see a 100% battery reading, and concluded that the laptop had sustained itself on power sucked out of the video cable. But my hosts assured me that this was a typical laptop failure and my battery was actually down. Indeed I've seen power meter nonsense on a very similar laptop, but I then continued to run on battery at a friends' house, where it gradually dropped to 90%, as if the "100%" were legit. Thought for today: (c69) (fancy_display:false,sqrt(2^(1/3)+5^(1/6)+5^(2/3)),%%=subst([2=t,t=2],substpart(expand(piece),denest_radicals(%%),1))) 2/3 1/6 1/3 (d69) sqrt(5 + 5 + 2 ) = 5/6 2/3 1/3 1/6 5 5 1/6 5 5 7/6 - ------- + ---- + 2 sqrt(5) + ------- + ---- + 2 sqrt(2) 1/6 sqrt(2) 1/6 2 2 ------------------------------------------------------- 3 (I tried to demo my favorite, from www.tweedledum.com/rwg/idents.htm, but in speeding up other cases, I made that one take forever.) --rwg squawky castrato: disharmonic anorchidism