26 Sep
2003
26 Sep
'03
8:28 p.m.
Steve Gray writes: << In "Kepler's Conjecture" by George Szpiro, it says that the naive angle-occupying calculation in 3-D gives an absolute upper bound of 14.9 spheres. This is just a solid-angle calculation. This should be easy to recompute by hand.
Right. One unit sphere tangent to another subtends a zone on the surface of the second consisting of all points within pi/6 of a single point. Since zone area is given by 2 pi R h = 2 pi h = 2 pi (1 - sqrt(3)/2), the upper bound based only on subtended spherical area is given by 4 pi divided by 2 pi (1 - sqrt(3)/2), or 4/(2-sqrt(3) = 14.928203... --Dan