Re: lattice packings with rigidity of the kissing balls



Chip Eastham wrote:
On Dec 3, 9:32 pm, David Bernier<david...@xxxxxxxxxxxx> wrote:
In the usual hexagonal packing of disks of the same diameter a plane,
the kissing number is six. This means that any disk is in contact
with six surrounding disks.

Also, if we only consider a disk and the six surrounding disks,
there isn't much choice in moving the six surrounding disks
while keeping them in contact with the central disk.
If the central disk is kept fixed in place, then the
six surrounding disks can be rotated simultaneously
into another position. But this (from initial to
final configuration) can be done by an orientation-preserving
isometry of the plane.

In three dimensions, I believe it's known and been proved what
the densest lattice packing of congruent balls can be or is.
From what I remember, one visualization is the way cannon balls
can be put on a plane surface with the centers in the shape
of a ZxZ lattice in the euclidean plane R^2, with the next
layer up fitting in the depressions between four lower balls,
and so on for higher layers. Then the kissing number for
this 3D lattice packing is 12. Taking a ball and the
12 balls in contact with it, I don't think it's rigid.

For example, a canon-ball is "surrounded" in a plane square packing
by 8 close neighbours, with 4 balls in contact and 4 others
at a distance of (2sqrt(2) - 2)R away, R being the radius.
With the four balls at (2sqrt(2) - 2)R away removed in the
3D packing, keeping just the 12 in contact, it seems to me
that the 12 in contact can be moved around, while keeping in
contact with the center ball, with the end result not
obtainable by an isometry of R^3 applied to the initial
configuration. So, in this sense, I believe the 3D
cannon-ball packing entails that a central ball and
the 12 kissing balls aren't in a rigid configuration.

The general question in higher dimensions is for which dimension
d of a euclidean space do there exist lattice packings such that:
- Taking a fixed ball A and
- adding all the balls that are in contact with A

results in a "figure" S that is rigid?

For d = 2, rigidity as above is possible.
For d = 3, I don't think rigidity is possible for any lattice packing.

David Bernier

There are actually two maximally dense lattice
packings of spheres. One of them has the ZxZ
planar sublattice that you describe, but both
of them have hexagonally packed sublattices.

One way to visualize both packings and why
they have equal density is to think about
layers of hexagonally packed spheres. From
the first layer to the second there is just
the centering of spheres of the second layer
upon "hollows" formed by three mutually
adjacent spheres in the first layer.

It is with the placement of a third hexagonal
layer atop the second that we have 2 choices.
Of course we want to place the spheres into
hollows of the second layer, but there are
two types: half center upon the spheres of
the first layer, and half upon hollows of the
first layer that were not "filled" by the
second layer.

The first of these is called hexagonal close
packing if that choice of layers is repeated
endlessly, and the second, as David describes
above, is called (face-centered) cubic close
packing. As both choices of where the third
layer goes give equal heights to the layers,
the densities are the same. Indeed the choice
of centering can be changed randomly with the
succeeding hexagonal layers without reducing
density, but to get a lattice packing we need
to be consistent with one choice or the other.

Gauss proved these are the regular sphere
packings of greatest density. More recently
Hales gave elaborate arguments to settle the
Kepler conjecture, that these are the sphere
packings of greatest density, regular or
irregular. These arguments are AFAIK still
under the refereeing process, lengthy in part
because of his use of computational methods.

It seems to me that these close packings are
rigid, in the sense that no sphere can move
while fixing another and maintaining contacts
except by rigid rotation. But perhaps I don't
know your meaning of rigidity?

regards, chip



Interesting explanation about the two maximum-density lattices
in R^3. The kind of rigidity I had in mind is best illustrated
by packing pennies in a plane. In the hexagonal-type lattice
packing in R^2, every penny is in contact with 6 pennies.

If one of them is P_0 and the six pennies in contact with P_0
are labeled P_1 though P_6, then with P_1 fixed or stuck to the
plane, and with P_1 through P_6 constrained to be in contact
with P_0, it's possible to rotate P_1 from the half-line
originating at the center of P_0 and passing through the center
of P_1 by any angle theta, from 0 to 2pi. If P_1 is rotated
clockwise by theta, and P_0 is fixed, and P_1 though P_6 are
constrained to stay in contact with P_0, then
P_2 through P_6 will have to rotate clockwise by theta
in a manner similar to P_1. The configuration I'm looking
at only involves seven pennies: any penny in the plane, plus
the six pennies that share a point with the central penny,
assuming pennies are thought of as closed unit disks.

In R^3, I take it from you that there are two maximal-density
packings of balls. I believe in either lattice packing,
a ball is in contact with 12 balls, thinking of closed
unit balls in R^3. The way you described the packings,
I think one could say that that for a fixed ball B, the ball
in the layer above , X, that is in contact with B could
be moved without moving the 11 others balls that touch B:
the configuration I consider is B together with the
twelve balls that touch B in either maximal-density
lattice packing.

I'm interested in lattice packings where a ball B and the balls
that touch B have the rigidity configuration I tried to
illustrate for R^2 and the hexagonal lattice packing.

For example, could it be that the Leech lattice packing
in R^24 is rigid : with B a ball in a Leech lattice
packing, . [Yes, I think from consulting Wikipedia... ]

Wikipedia:
<< This arrangement of 196560 unit balls centered about another unit ball is so efficient that there is no room to move any of the balls; >>

from:
http://en.wikipedia.org/wiki/Leech_lattice

David

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Key fingerprint = D85C 4B36 AF9D 6838 CC64 20DF CF37 7BEF 6537 21FF
uid David Bernier (Biggy) <david250@xxxxxxxxxxxx>
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