Tony and Dave
There HAS to be a flaw in this logic. Although Tony has expressed that size
DOES matter. Let me explain.
I think the flaw is to assume perfect packing that yields the magic 26% open
space regardless of the size of the spheres. That level of packing would only
occur if the dimensions of the space were an INTEGRAL multiple of the diameters
of the sphere, and if there were no wasted space above the spheres!
Think of the case of a POWDER - essentially very very tiny spheres, packed into
a 100x100 format (i.e. their diameters are 1/100 of the dimension of the space)
versus large spheres in a 1x1 format. CLEARLY you're going to get more stuff into
the space with the powder.
In other words, in a 1x1 format, the "empty space" is actually 1.00 - 4.19/8.00 =
.47625 or 47% empty space! The 26% empty space is a BOUNDARY CONDITION of maximally
In any case, it is subject to experiment to determine whether this is true, or not.
Or to put it another way, the proportion of space occupied by spheres, even in
the closest packing, is independent of the size of the spheres. Of course, if you
have a range of sizes of spheres, the little ones pack between the big ones. But
if they are all the same size, using smaller ones changes nothing.