The volume formula for the perpendicular intersection of two cylinders
of radius r can be found by replacing pi by 4 in the volume of the
inscribed sphere. It is said that Archimedes found this thirteen
centuries before Newton's calculus. Did he also know that the same
4/pi trick gives the surface area? If not, perhaps he noticed the
universal prismatoid volume formula
A + 4 A + A
t m b
V = h --------------,
6
(= one panel of Simpson's rule, with A:=area, t:=top, m:=middle,
b:=bottom, h=height), which works on all the elementary solids, and
deduced that it also works for the "biroller". (Is there a classical
name for this solid?)
I've said this before, but a fairly dense, fairly hard biroller
(say, lathed from 2" aluminum bar stock) makes an intriguing chaotic
toy, when rolled diagonally down a shallow, inclined trough (like a
bathtub, but longer and shallower). The roller oscillates with
gradually decaying amplitude until a downswing pauses on one of the
two poles, whereupon it rocks with renewed amplitude, downslope on a
perpendicular arc, taking a completely unpredictable time to reach the
bottom. This was shown to me by an adolescent David Silver ca. 1970.
--rwg
PS, a Macsyma command to draw a biroller is
block([plotnum1:2,f],[sin(t),sin(t),cos(t)],f:(1-u)*%%+u*[-1,1,1]*%%,
makelist(f:part(f*[-1,1,1],[2,1,3]),k,1,4),plotsurf(%%,t,0,%pi,u,0,1))
