Packing unit spheres in cubes: new results

[David W. Cantrell]

Eckard Specht has recently added to Packomania <http://www.packomania.com/>
a new section about packing equal spheres in cubes.

And I have now found four improvements:
N = 47, 55, 56 and 57 unit spheres can be packed in cubes of side lengths
7.14495694718505884469538734132,
7.63741768887083424431741951543,
7.65575717475793069412004927357 and
7.65681613178383380764819773560, respectively.
These improved packings will probably be shown at Packomania soon.

David W. Cantrell

[David W. Cantrell]

Here are a few more improvements:

N = 65 and 68 unit spheres can be packed in cubes of side lengths
7.92522084972161214760499929193 and 8.10369112913570676974347890930, resp.

N = 180 unit spheres can be packed in a cube of side length
2 + 5/13 (20 + sqrt(10)). This is a 3D analog of the column-shift packing
of 30 unit circles in a square.

N = 416 unit spheres can be packed in a cube of side length
2 + 168/sqrt(193). This is a 3D analog of the grid packing of 52 unit
circles in a square. Similarly, N = 567 unit spheres can be packed in a
cube of side length 2 + 208/sqrt(233).

David W. Cantrell

[David W. Cantrell]

Packomania now shows several other new grid packings, as well as many other
improved packings by Eckard Specht, by the program Pack'n'tile and by me.

New packings found by me which have yet to appear at Packomania include the
following:

N = 87, side length = 2 + 14/3 sqrt(2)
Highly symmetric! Precise coordinates of the centers of the 87 unit spheres
are given below my signature.

N = 98, side length = 8.9670822119424704015935524436153786878471791...
N = 99, side length = 8.9776912998071394503672341290673989694456976...
Both of the above packings have one plane of symmetry.

David W. Cantrell

----------------------------------------------------------------

{0, 0, -2*Sqrt[2]}, {0, 0, 2*Sqrt[2]}, {0, -2*Sqrt[2], 0},
{0, -Sqrt[2], -Sqrt[2]}, {0, -Sqrt[2], Sqrt[2]},
{0, Sqrt[2], -Sqrt[2]}, {0, Sqrt[2], Sqrt[2]},
{0, 2*Sqrt[2], 0}, {(-7*Sqrt[2])/3, (-7*Sqrt[2])/3,
(-4*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (-7*Sqrt[2])/3,
(2*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (-4*Sqrt[2])/3,
(-7*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (-4*Sqrt[2])/3,
-Sqrt[2]/3}, {(-7*Sqrt[2])/3, (-2*Sqrt[2])/3,
(7*Sqrt[2])/3}, {(-7*Sqrt[2])/3, -Sqrt[2]/3,
(-4*Sqrt[2])/3}, {(-7*Sqrt[2])/3, Sqrt[2]/3,
(4*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (2*Sqrt[2])/3,
(-7*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (4*Sqrt[2])/3,
Sqrt[2]/3}, {(-7*Sqrt[2])/3, (4*Sqrt[2])/3,
(7*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (7*Sqrt[2])/3,
(-2*Sqrt[2])/3}, {(-7*Sqrt[2])/3, (7*Sqrt[2])/3,
(4*Sqrt[2])/3}, {-2*Sqrt[2], 0, 0},
{-2*Sqrt[2], -Sqrt[2], Sqrt[2]}, {-2*Sqrt[2], Sqrt[2],
-Sqrt[2]}, {(-4*Sqrt[2])/3, (-7*Sqrt[2])/3,
(-7*Sqrt[2])/3}, {(-4*Sqrt[2])/3, (-7*Sqrt[2])/3,
-Sqrt[2]/3}, {(-4*Sqrt[2])/3, (-4*Sqrt[2])/3,
(-4*Sqrt[2])/3}, {(-4*Sqrt[2])/3, -Sqrt[2]/3,
(-7*Sqrt[2])/3}, {(-4*Sqrt[2])/3, Sqrt[2]/3,
(7*Sqrt[2])/3}, {(-4*Sqrt[2])/3, (4*Sqrt[2])/3,
(4*Sqrt[2])/3}, {(-4*Sqrt[2])/3, (7*Sqrt[2])/3,
Sqrt[2]/3}, {(-4*Sqrt[2])/3, (7*Sqrt[2])/3,
(7*Sqrt[2])/3}, {-Sqrt[2], 0, -Sqrt[2]},
{-Sqrt[2], 0, Sqrt[2]}, {-Sqrt[2], -2*Sqrt[2], Sqrt[2]},
{-Sqrt[2], -Sqrt[2], 0}, {-Sqrt[2], -Sqrt[2], 2*Sqrt[2]},
{-Sqrt[2], Sqrt[2], 0}, {-Sqrt[2], Sqrt[2], -2*Sqrt[2]},
{-Sqrt[2], 2*Sqrt[2], -Sqrt[2]}, {(-2*Sqrt[2])/3,
(-7*Sqrt[2])/3, (7*Sqrt[2])/3}, {(-2*Sqrt[2])/3,
(7*Sqrt[2])/3, (-7*Sqrt[2])/3},
{-Sqrt[2]/3, (-7*Sqrt[2])/3, (-4*Sqrt[2])/3},
{-Sqrt[2]/3, (-4*Sqrt[2])/3, (-7*Sqrt[2])/3},
{-Sqrt[2]/3, (4*Sqrt[2])/3, (7*Sqrt[2])/3},
{-Sqrt[2]/3, (7*Sqrt[2])/3, (4*Sqrt[2])/3},
{Sqrt[2]/3, (-7*Sqrt[2])/3, (4*Sqrt[2])/3},
{Sqrt[2]/3, (-4*Sqrt[2])/3, (7*Sqrt[2])/3},
{Sqrt[2]/3, (4*Sqrt[2])/3, (-7*Sqrt[2])/3},
{Sqrt[2]/3, (7*Sqrt[2])/3, (-4*Sqrt[2])/3},
{(2*Sqrt[2])/3, (-7*Sqrt[2])/3, (-7*Sqrt[2])/3},
{(2*Sqrt[2])/3, (7*Sqrt[2])/3, (7*Sqrt[2])/3},
{Sqrt[2], 0, -Sqrt[2]}, {Sqrt[2], 0, Sqrt[2]},
{Sqrt[2], -2*Sqrt[2], -Sqrt[2]}, {Sqrt[2], -Sqrt[2], 0},
{Sqrt[2], -Sqrt[2], -2*Sqrt[2]}, {Sqrt[2], Sqrt[2], 0},
{Sqrt[2], Sqrt[2], 2*Sqrt[2]}, {Sqrt[2], 2*Sqrt[2],
Sqrt[2]}, {(4*Sqrt[2])/3, (-7*Sqrt[2])/3, Sqrt[2]/3},
{(4*Sqrt[2])/3, (-7*Sqrt[2])/3, (7*Sqrt[2])/3},
{(4*Sqrt[2])/3, (-4*Sqrt[2])/3, (4*Sqrt[2])/3},
{(4*Sqrt[2])/3, -Sqrt[2]/3, (7*Sqrt[2])/3},
{(4*Sqrt[2])/3, Sqrt[2]/3, (-7*Sqrt[2])/3},
{(4*Sqrt[2])/3, (4*Sqrt[2])/3, (-4*Sqrt[2])/3},
{(4*Sqrt[2])/3, (7*Sqrt[2])/3, (-7*Sqrt[2])/3},
{(4*Sqrt[2])/3, (7*Sqrt[2])/3, -Sqrt[2]/3},
{2*Sqrt[2], 0, 0}, {2*Sqrt[2], -Sqrt[2], -Sqrt[2]},
{2*Sqrt[2], Sqrt[2], Sqrt[2]}, {(7*Sqrt[2])/3,
(-7*Sqrt[2])/3, (-2*Sqrt[2])/3},
{(7*Sqrt[2])/3, (-7*Sqrt[2])/3, (4*Sqrt[2])/3},
{(7*Sqrt[2])/3, (-4*Sqrt[2])/3, Sqrt[2]/3},
{(7*Sqrt[2])/3, (-4*Sqrt[2])/3, (7*Sqrt[2])/3},
{(7*Sqrt[2])/3, (-2*Sqrt[2])/3, (-7*Sqrt[2])/3},
{(7*Sqrt[2])/3, -Sqrt[2]/3, (4*Sqrt[2])/3},
{(7*Sqrt[2])/3, Sqrt[2]/3, (-4*Sqrt[2])/3},
{(7*Sqrt[2])/3, (2*Sqrt[2])/3, (7*Sqrt[2])/3},
{(7*Sqrt[2])/3, (4*Sqrt[2])/3, (-7*Sqrt[2])/3},
{(7*Sqrt[2])/3, (4*Sqrt[2])/3, -Sqrt[2]/3},
{(7*Sqrt[2])/3, (7*Sqrt[2])/3, (-4*Sqrt[2])/3},
{(7*Sqrt[2])/3, (7*Sqrt[2])/3, (2*Sqrt[2])/3}, {0, 0, 0},
{-1/100 + (7*Sqrt[2])/3, 1/100 - (7*Sqrt[2])/3,
1/100 - (7*Sqrt[2])/3}, {1/100 - (7*Sqrt[2])/3,
-1/100 + (7*Sqrt[2])/3, 1/100 - (7*Sqrt[2])/3},
{-1/100 + (7*Sqrt[2])/3, -1/100 + (7*Sqrt[2])/3,
-1/100 + (7*Sqrt[2])/3}, {1/100 - (7*Sqrt[2])/3,
1/100 - (7*Sqrt[2])/3, -1/100 + (7*Sqrt[2])/3}

[David W. Cantrell]

David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> Packomania now shows several other new grid packings, as well as many
> other improved packings by Eckard Specht, by the program Pack'n'tile and
> by me.

See <http://hydra.nat.uni-magdeburg.de/packing/scu/>.

> New packings found by me which have yet to appear at Packomania include
> the following:
>
> N = 87, side length = 2 + 14/3 sqrt(2)
> Highly symmetric! Precise coordinates of the centers of the 87 unit
> spheres are given below my signature.

That packing for N = 87 appeared at Packomania on Feb. 29, 2012, attributed
to Huang and Yu. But it was presented here slightly earlier (Feb. 27) by me.

That packing and several other known packings are members of a particularly
nice family of packings. In that family, spheres are grouped in pyramids in
four corners of the cube and in a tetrahedron in the middle of the cube. It
is perhaps easiest to break the family into two subfamilies, depending on
whether or not one of the spheres is at the center of the cube:

Letting n be a positive integer, then

N = (4 n^2 + 6 n + 1)(n + 1) unit spheres
can be packed in a cube of edge length (2 n + 2/3) sqrt(2) + 2

and

N = 4 n^2 (n + 1) unit spheres
can be packed in a cube of edge length (2 n - 1/3) sqrt(2) + 2.

Specific new packings in this family which will now be sent to Packomania
are those for N = 220, 445, 786 and 1008. (Precise coordinates of the
centers of the spheres are available on request.)

[David W. Cantrell]

1. The family mentioned previously is discussed further and a picture of
its structure is given.
2. A new family of packings is described.

Taking the previous and new families together with the family of cubic
close packings, we now have packings for cubes of edge length
k/3 sqrt(2) + 2 for all positive integer k.

---------------------------------------------------------------------------

David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> > Packomania now shows several other new grid packings, as well as many
> > other improved packings by Eckard Specht, by the program Pack'n'tile
> > and by me.
>
> See <http://hydra.nat.uni-magdeburg.de/packing/scu/>.
>
> > New packings found by me which have yet to appear at Packomania include
> > the following:
> >
> > N = 87, side length = 2 + 14/3 sqrt(2)
> > Highly symmetric! Precise coordinates of the centers of the 87 unit
> > spheres are given below my signature.
>
> That packing for N = 87 appeared at Packomania on Feb. 29, 2012,
> attributed to Huang and Yu. But it was presented here slightly earlier
> (Feb. 27) by me.
>
> That packing and several other known packings are members of a
> particularly nice family of packings. In that family, spheres are grouped
> in pyramids in four corners of the cube and in a tetrahedron in the
> middle of the cube.

Here's a picture of the structure. The cyan line segments, of length 2,
indicate tangencies between unit spheres. (By the way, the cube shown is
automatically provided by the graphing program; it is slightly larger than
the cube which encloses the sphere centers.) So that the picture would not
be too cluttered to see the structure, the picture shows only 220 sphere
centers in a region which is adequate to contain 445 sphere centers. It
should then be easy to visualize how, by adding layers of spheres to the
four corner pyramids and to the central tetrahedron, we can go from a
packing for N = 220 to one for N = 445:

<http://i403.photobucket.com/albums/pp113/DWCantrell_photos/sph220cu.gif>

> It is perhaps easiest to break the family into two
> subfamilies, depending on whether or not one of the spheres is at the
> center of the cube:
>
> Letting n be a positive integer, then
>
> N = (4 n^2 + 6 n + 1)(n + 1) unit spheres
> can be packed in a cube of edge length (2 n + 2/3) sqrt(2) + 2

In this first subfamily, which has a sphere at the center of the cube, no
sphere in the pyramidal groups is tangent to three faces of the cube.

> and
>
> N = 4 n^2 (n + 1) unit spheres
> can be packed in a cube of edge length (2 n - 1/3) sqrt(2) + 2.

In this second subfamily, which does not have a sphere at the center of the
cube, the sphere at the vertex of each pyramidal group is tangent to three
faces of the cube.

All packings in this family have four rattlers.

Currently, it seems possible that all members of the first subfamily
(N = 22, 87, 220, 445, 786, ...) are optimal packings. But that is not true
for the second subfamily (N = 8, 48, 144, 320, 600, 1008, ...) since better
packings are known for N = 8 and 48.

---------------------------------------------------------------------------

A new family

Imagine that we have just two pyramidal groups of spheres in opposite
corners of the cube. We may then add more layers of spheres to the two
groups. But eventually the groups will no longer be pyramids because later
layers will need to be curtailed so that all spheres continue to lie within
the cube. We can continue to add curtailed layers until the spheres of one
group finally touch those of the other group. A packing for N = 395
constructed in this way is shown at
<http://i403.photobucket.com/albums/pp113/DWCantrell_photos/sph395cu.gif>.

Letting n be a positive integer:

i) N = (8 n^2 + 7 n + 2)(n + 1)/2 unit spheres
can be packed in a cube of edge length (2 n + 1/3) sqrt(2) + 2.

In this first subfamily, there is a sphere tangent to three faces of the
cube at one corner, but not in the opposite corner, as in the packing shown
for N = 395. Currently, it seems possible that all members of this
subfamily (N = 17, 72, 190, 395, 711, ...) are optimal packings.

ii) N = (8 n^3 + 3 n^2 + ((-1)^n - 1)/2)/2 unit spheres
can be packed in a cube of edge length (2 n - 2/3) sqrt(2) + 2.
[Alternatively, N = (8 n^3 + 3 n^2)/2 for even n
and N = (8 n^3 + 3 n^2 - 1)/2 for odd n.]

In this second subfamily, at opposite corners of the cube, there two
spheres tangent to three faces, as in the packing for N = 121, already
shown at <http://hydra.nat.uni-magdeburg.de/packing/scu/scu121.html>. Not
all members of this subfamily (N = 5, 38, 121, 280, 537, 918, ...) are
optimal since better packings are known for N = 5 and 38.

For this new family, the four packings with 300 < N < 1000 will be sent to

[David W. Cantrell]

1. The family mentioned previously is discussed further and a picture
of its structure is given.
2. A new family of packings is described.

Taking the previous and new families together with the family of
cubic close packings, we now have packings for cubes of edge length
k/3 sqrt(2) + 2 for all positive integer k.

(This post contains an important correction to a post made earlier today,
now supposedly superseded.)

---------------------------------------------------------------------------

David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> > Packomania now shows several other new grid packings, as well as many
> > other improved packings by Eckard Specht, by the program Pack'n'tile
> > and by me.
>
> See <http://hydra.nat.uni-magdeburg.de/packing/scu/>.
>
> > New packings found by me which have yet to appear at Packomania include
> > the following:
> >
> > N = 87, side length = 2 + 14/3 sqrt(2)
> > Highly symmetric! Precise coordinates of the centers of the 87 unit
> > spheres are given below my signature.
>
> That packing for N = 87 appeared at Packomania on Feb. 29, 2012,
> attributed to Huang and Yu. But it was presented here slightly earlier
> (Feb. 27) by me.
>
> That packing and several other known packings are members of a
> particularly nice family of packings. In that family, spheres are grouped
> in pyramids in four corners of the cube and in a tetrahedron in the
> middle of the cube.

Here's a picture of the structure. The cyan line segments, of length 2,
indicate tangencies between unit spheres. (By the way, the cube shown is
automatically provided by the graphing program; it is slightly larger than
the cube which encloses the sphere centers.) So that the picture would not
be too cluttered to see the structure, the picture shows only 220 sphere
centers in a region which is adequate to contain 445 sphere centers. It
should then be easy to visualize how, by adding layers of spheres to the
four corner pyramids and to the central tetrahedron, we can go from a
packing for N = 220 to one for N = 445:

<http://i403.photobucket.com/albums/pp113/DWCantrell_photos/sph220cu.gif>

> It is perhaps easiest to break the family into two
> subfamilies, depending on whether or not one of the spheres is at the
> center of the cube:
>
> Letting n be a positive integer, then
>
> N = (4 n^2 + 6 n + 1)(n + 1) unit spheres
> can be packed in a cube of edge length (2 n + 2/3) sqrt(2) + 2

In this first subfamily, which has a sphere at the center of the cube, no
sphere in the pyramidal groups is tangent to three faces of the cube.

> and
>
> N = 4 n^2 (n + 1) unit spheres
> can be packed in a cube of edge length (2 n - 1/3) sqrt(2) + 2.

ii) N = (8 n^3 + 3 n^2 + (1 - (-1)^n)/2)/2 unit spheres

can be packed in a cube of edge length (2 n - 2/3) sqrt(2) + 2.
[Alternatively, N = (8 n^3 + 3 n^2)/2 for even n

and N = (8 n^3 + 3 n^2 + 1)/2 for odd n.]

In this second subfamily, at two opposite corners of the cube, when n is
even, there are spheres tangent to three faces and, when n is odd, there is
no sphere tangent to three faces. Not all members of this subfamily (N = 6,
38, 122, 280, 538, 918, ...) are optimal since a better packing is known
for N = 38.

For this new family, packings with N < 1000 will be sent to Packomania
soon.

David W. Cantrell

[1treePetrifiedForestLane]

what are teh results for packing a sphere ... lunes?

[David W. Cantrell]

Our third (and perhaps final) new family of packings is described below.
It gives packings of unit spheres in cubes of edge length k sqrt(2) + 4 for
positive integer k.

(In a post soon to appear, we will discuss density records and compare
families of packings.)

-------------------------------------------------------------------------

David W. Cantrell <DWCan...@sigmaxi.net> wrote:
> 1. The family mentioned previously is discussed further and a picture
> of its structure is given.
> 2. A new family of packings is described.
>
> Taking the previous and new families together with the family of
> cubic close packings, we now have packings for cubes of edge length
> k/3 sqrt(2) + 2 for all positive integer k.
>

-------------------------------------------------------------------------

A third new family

For positive integer n, N = n^2 (n + 1)/2 unit spheres
can be packed in a cube of edge length (n - 2) sqrt(2) + 4.

The structure of packings in this family is similar to that of our first
new family, in that the central grouping of spheres is a tetrahedron and
the four corner groupings are (with one exception to be discussed)
pyramids. But in the first new family, all four pyramidal groups either had
or did not have a sphere at the vertex of the pyramid tangent to three
faces of the cube; in the third new family, two pyramidal groups do and the
other two do not have a sphere tangent to three faces of the cube. It will
be easiest to describe the structure more fully if we consider even and odd
n separately.

----------

Even n

As an example, with n = 8, consider Eckard Specht's packing for N = 288
unit spheres in a cube of edge length 6 sqrt(2) + 4, shown at
<http://i403.photobucket.com/albums/pp113/DWCantrell_photos/sph288cu.gif>.

There are two incomplete pyramidal groups in the upper left and right
corners of the cube. At the vertices of those pyramids, there are no
spheres which are tangent to three faces of the cube. Each of those
incomplete pyramidal groups has n/2 layers of spheres. But the innermost
layer of each is incomplete, missing a row of spheres at the top of the
cube. (If both of those pyramids had been completed, the spheres in the two
upper rows would have been too close together.) But there is room for one
row of spheres (rattlers) near the top of the cube.

And there are two (complete) pyramidal groups in the lower front and back
corners of the cube. At the vertex of each of those pyramids, there is a
sphere tangent to three faces of the cube. Each of those pyramidal groups
also has n/2 layers of spheres. And of course the tetrahedral grouping of
spheres fills in the central part of the cube, touching all four pyramidal
groups.

----------

Odd n

As an example, with n = 9, consider the new packing for N = 405 unit
spheres in a cube of edge length 7 sqrt(2) + 4, shown at
<http://i403.photobucket.com/albums/pp113/DWCantrell_photos/sph405cu.gif>.

All four pyramidal groups are complete. At the vertices of those in the
upper left and right corners, there are no spheres tangent to three faces
of the cube; each of those pyramidal groups has (n - 1)/2 layers of
spheres. At the vertices of the pyramidal groups in the lower front and
back corners, there are spheres tangent to three faces of the cube; one of
those pyramidal groups has (n - 1)/2 layers of spheres and the other has
(n + 1)/2 layers. (Thus, the packings for odd n are not as highly symmetric
as those for even n.) And the tetrahedral grouping of spheres fills in the
central part of the cube, touching all four pyramidal groups.

----------

As the soon-to-appear comparison of packing families will show, packings in
this third new family are not quite as good, for large N, as those in the
second new family. For the first six values of N (1, 6, 18, 40, 75 and
126), packings in this family are not optimal, but they are close for the
latter two. For N = 75, the best packing known has edge length e =
8.24256..., while the packing in this family has e = 3 sqrt(2) + 4 =
8.24264... And for N = 126, the best packing known has e =
9.656854249449..., while the packing in this family has e = 4 sqrt(2) + 4 =
9.656854249492... For larger N (196, 288, 405, 550, 726, 936, ...),
packings in this family are currently the best ones known and are perhaps
optimal. (If they are not optimal, improvements will likely be very small
and difficult to find.)

For N < 1500, except for N = 405 and 1470, which will be sent to Packomania
soon, other possibly optimal packings in this family already appear there.

David W. Cantrell