# Re: lattice packings with rigidity of the kissing balls

*From*: David Bernier <david250@xxxxxxxxxxxx>*Date*: Sat, 04 Dec 2010 11:03:12 -0500

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

--

$ gpg --fingerprint david250@xxxxxxxxxxxx

pub 2048D/653721FF 2010-09-16

Key fingerprint = D85C 4B36 AF9D 6838 CC64 20DF CF37 7BEF 6537 21FF

uid David Bernier (Biggy) <david250@xxxxxxxxxxxx>

.

**Follow-Ups**:**Re: lattice packings with rigidity of the kissing balls***From:*Chip Eastham

**References**:**lattice packings with rigidity of the kissing balls***From:*David Bernier

**Re: lattice packings with rigidity of the kissing balls***From:*Chip Eastham

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