BBC warning: this post may contain "flash photography."
Let me get going on this Step-Function. It is a step function because
the x is a Real number indicating the length of the cube-container
and the y component is the total number of spheres packed inside
that container. Now if Kepler Packing was only involving Hexagonal
Close Packing then this function would probably have some resemblence
to that of f(x) = x^2. But since it has two choices of maximum
packing of the Corner-Edge-HCP and the HCP-alone then the
function is going to be a Step-function, similar to making x^2 a
step function.
And a step-function is in accord with Kepler Packing being
discrete-geometry.
So let me speak of what I see in my imagination of this function
and then go and experiment with the marbles to see if the
imagination is accurate.
Let me use the case of N= 11 of a cube 11units long. Since I
have the marbles of a 11 stacking.
Let me make this line as indicating the top of the cube
and the topmost two layers of spheres
_________
OOOOOO
OOOOOO
Now in the 11cube there is some headway of a space between
the top of the cube and the last layer. For my marbles have
an average diameter of 160mm/11 = 14.5mm and (my post
#664) the headway is 160mm - 147mm = 13mm
So there is not enough room for another layer but plenty of
room to scoot the entire upper layer to the rightmost edge.
And since the top layer is no longer nested in the hollows that
elevates the toplayer and removes some of that headway.
Now in the cubes going larger from N=11 to say 11.1 then
11.2 then 11.3, somewhere along this increase in cube size
where the headway is so large that entire new upper layer
can be rested into the hollows of the below layer.
So from say N= 11 to about N = 11.4 we have Corner-Edge-
HCP.
Then from N= 11.4 until about 11.6 or 11.7 we have a headway
not large enough to scoot over the entire upper layer so that
we return to the Corner-Edge-HCP. And so from N= 11.4
to N= 11.7 we have HCP-alone. And finally in the interval
N=11 to N=12 we return to the Corner-Edge-HCP from
11.7 to past N=12, and then the cycle repeats. So we cycle
from Corner-Edge-HCP to HCP-alone.
Let me try to summarize and remember what happens.
From 11 to 11.4 we have Corner-Edge-HCP
Then from 11.4 to 11.7 we have HCP-alone
Then from 11.7 to 2.2 we have Corner-Edge-HCP
and cycle through these from one interval to the next.
So let me see if the marbles conform to the above
mind model.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
