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Are Denser Packings of Spheres than Closest Packings Possible How Many Closest Packings of Spheres Exist.

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Are Denser Packings of Spheres than Closest Packings Possible? How Many Closest
Packings of Spheres Exist?
By Ulrich Miiller*
In 1611 Johannes Kepler wrote a humorous 24-page epistle
which was a New Year’s present to his benefactor in Prague,
the Imperial Councilor Wackher von Wackenfels.[’I Here he
proposed an explanation of the hexagonal shape of snowflakes: Water vapor, once it feels the penetrating coldness,
freezes into little spheres with some definite size which he calls
vapor spheres (sphaere vapidae). The vapor spheres in contact
with one another adopt the arrangement that we nowadays
call cubic closest packing of spheres and which we know to be
the structural principle of many metals and of the solid noble
gases. For this packing Kepler asserted, “It is the tightest
possible; in no other arrangement can more spheres be accommodated in the same vessel.”[21The term closest packing
of spheres shows that chemists, physicists, and crystallographers have long intuitively accepted Kepler’s assertion; however, it was not proven mathematically until 1991 by M! Y.
For periodically ordered packings of spheres, i.e. lattice
packings, Gauss furnished the proof of the highest packing
density. although he did not mention packings of spheres at
However, he introduced the concept of a lattice, and by
relating it to the Seeber’s theorems[5] on number theory he
stated: the volume V of the (primitive) unit cell of a lattice
will always satisfy the condition V@ 2 abc if the face and
body diagonals of the cell are not shorter than the lattice
parameters u, h, and c. The upper limit of the space filling of
a lattice packing of equal spheres follows from this condition
as was shown by Minkowski.[61The primitive unit cell of the
cubic closest packing of spheres fulfilling Gauss’s condition
is a rhombohedra1 cell with the cell edges a = b = c = 2r and
the volume V = $a3@ ( r = radius of a sphere). This volume is exactly equal to the limiting value of the Gauss condition: V l / z = ( $ a 3 f i ) f i 2 a3 (it also is identical for any
other closest packing of spheres). A crystal lattice having a
smaller unit cell volume, i.e. with greater packing density, is
therefore not possible for a packing of equal spheres. This
statement, however, is valid only when there is a crystal
Just as chemists presently have caught the buckyball fever,
crystallographers are fiercily discussing the topic of quasi
crystals. Quasi crystals have symmetry (e.g. fivefold rotation
axes or icosahedral symmetry), and there is a near order
around the atoms, but they have no three-dimensional periodical order. Could a quasi-crystalline packing of spheres exist
having a higher degree of space filling than a cubic closest
packing of spheres? Since the twelve spheres surrounding a
central sphere of the same size in an icosahedral arrangement
are not in contact with each other, a larger space filling seems
possible at first (the coordination number in a closest packing
of spheres is 12). Mathematicians have repeatedly furnished
[*] Prof. Dr. U. Muller
FB Chemie der Universitit
Hans-Meerwein-Strasse, D-W-3550 Marburg (FRG)
Angrn. Clirrn. lnt. Ed. Engl. 31 (1992) N o . 6
proofs with upper limits for the space filling, with values that
have been decreasing over the years in the third or fourth
decimal place, e.g. 0.77964 (1958), 0.77844 (1986), and finally
0.77836 (1988). There have also been repeated attempts to
design nonperiodical packings of spheres, for instance with
fivefold symmetry[’] or consisting of shells with icosahedral
symmetry,[’] but not even the space filling of a closest packing
of spheres was ever achieved.
The central idea of Hsiang’s proof concerns the domains of
influence of the spheres (also called the Wirkungsbereich,
Voronoi domain, Dirichlet domain, or Wigner-Seitz cell). The
domain of influence of a sphere is the polyheron made up of
the set of planes perpendicular to the lines that connect the
sphere under consideration to the surrounding spheres and
passing through the midpoints of these lines. (For spheres in
contact with each other these are the tangent planes between
the spheres.) Kepler described the domain of influence of a
sphere in the cubic closest packing by considering the seeds
in a pomegranate. Initially the seeds are round, but later as
the rind hardens and the seeds continue to grow, they become tightly packed and squeezed together, filling the remaining space and finally taking the shape of rhombic dodecahedra (Fig. 1). In an arbitrary packing of spheres the
domains of influence fill the space completely, so that the
space filling results from the ratio of the sphere volume to the
mean volume of all domains of influence. The domain of
influence of the cubic closest packing of spheres, the rhombic
dodecahedron, has a volume of 4l,h when the sphere radius
is r = I . With the sphere volume f n we thus calculate the
well-known degree of space filling of the cubic closest packing of spheres to be 47c/(41/2) = kn@ = 0.7404805.
The smallest possible local domain of influence of a single
sphere results when it is coordinated i c ~ s a h e d r a l l y ;this
domain of influence is a pentagonal dodecahedron (Fig. 1).
It has a volume of 101/2(65 - 291/J), which yields a local
space filling of 0.7546974. This value had been suspected to
be the upper limit for sphere packings but was never proven.”]
Since space cannot be completely filled by pentagonal dodecahedra, other spheres must have different domains of influence. What is the minimum mean volume of all domains of
Fig. 1 . Rhombic dodecahedron (left) and pentagonal dodecahedron (right)
enclosing spheres of equal size.
VCH K~rlagsgesellsrhafimbH, W-6940 Weinheim. 1992
OS70-0833/92/0606-0727 $3.50+.2S/O
Tahle 1. Number of possible stacking variants of closest packings of spheres depending on the space group type 1151. N
1 [dl
2 040
65 268
4 192200
65 240
57 208
488 667 525 580
of layers per packet
1 [bl
59 290
159 139498
= number
[a] Hexagonal closest packing. [b] Cubic closest packing. space group F m h
In addressing this question Hsiang considered a second
shell of spheres, consisting of 42 to 44 spheres around the 12
spheres of the first shell; including the central sphere this
amounts to a total of 55 to 57 spheres. His impressive goal
was to analyze all possible configurations and to derive the
domains of influence of the spheres of the first shell. Most
configurations could be eliminated because the volumes are
too large; for other configurations it had to be shown that
they are close to certain key configurations. The distinction
of these cases makes up the greater part of the proof which
extends over more than 150 pages. First a new method for
estimating the volume of local domains of influence was used
to prove that the global space filling cannot exceed the local
space filling in a pentagonal dodecahedron. Then after introducing the concept of semiglobal space-filling, he derived the
maximum global space-filling; it is 0.7404805.
Hsiang first circulated his proof among mathematicians
who now have the opportunity to check its correctness. In
other circles his work became known by two communications," 0 %111
Another problem that had never been solved was the question of the possible number of different closest packings of
spheres that all have the same space filling as cubic closest
packing. For stacks of hexagonal layers of spheres, any
stacking sequence of layers with the positions A , B, and C is
possible as long as no two layers of the same position are
adjacent; therefore, the number of stacking variants is infinite. But how many arrangements are possible for periodical stackings in which a packet of a given number of layers
is repeated? First of all, we have to make a clarifying statement: if we take the cubic closest packing of spheres and
consider a stacking direction inclined towards the layer plane
A + B + C -+ A ..., then we have a translationary equivalent
layer after one layer (one layer per packet). However, if we
only consider a stacking direction perpendicular to the layers, the packet is repeated only after three layers,
(ABC) (ABC) ... An inclined stacking of a packet having N
layers will always result in a rhombohedra1 structure which
can be described in a hexagonal setting with packets consisting of 3N layers. In the following we will only consider the
perpendicular stacking direction.
For a small number of layers per packet the possible stacking sequences can be written down and
method for calculating the number of possibilities for an
$3 VCH
Verlugsgesdl.schu/r mbH, W-6940 Wernhcim, 1992
arbitrary number of layers per packet was first demonstrated
in 1981 by McLarnan.[l5I This approach is based on Polya's
theorem,[l6](which can also be used to calculate the number
of isomers of organic compounds). If, in addition, some
group theory and White's extension of Polya's theorem are
included,[", 'I the number of stacking variants with a given
symmetry (space group) can also be calculated. Table 1 lists
some of the numbers calcuhted by McLarnan. The table
illustrates why it is a significant progress to be able to calculate the numbers with dependence on symmetry: The largest
numbers result for low symmetries, whereas in the higher
symmetry space group types P6,lmmc and R3m the numbers
are relatively small. Chemically reasonable crystal structures
are usually only those with higher symmetry. The method
described by McLarnan["] can also be adapted for the calculation of the number of crystal structure types possible with
particular space groups for any kind of compound, provided
that some definite packing principle can be stated."
I thank Prof. Dr. E. Krajft for helping me locate Kepler's
German version: Angew. Chrm. 1992, 104. 744
[l] J. Kepler, Srrena seu de nive se.xungulu, G. Tampach. Frankfurt. 1611.
Reprinted in: Johannes Keplrr -gesummelie Werke(Eds.:W. van Dyck. M.
Caspar, F. Hammer), Vol. IV, C. H. Beck, Miinchen, 1941, p. 261-280.
English translation: On the SiY-Cornered Snowjluke, Oxford University
Press, 1966.
[2] "Coaptatio fiet arctissima: ut nullo praetera ordine plures glohuli in idem
vas compingi queant."
[3] W. Y Hsiang, Reports PAM-530 and PAM-535. Center for Pure and Applied Mathematics, University of California, Berkeley, 1991. Bull. Bra:&
iun Murh. Soc., in press.
[4] C. F. Gauss, Wcrke 11. Kiinigliche Gesellschaft der Wissenschaften.
Gottingen, 1876. pp. 388-~196.
(51 L. A. Seeber. Unrrr.~uchrcngenuber die Eigenschu/ten der posiriveii rernui-en
yuadrurrschen Formen, Freihurg, 1831.
[6] H. Minkowski, Nurhr. Ges. W i v . Garringen Murh. P h u . KI. 1904, 311.
[7] B. G. Bagley, Narure 1965. 208. 674; ;bid. 1970, 225. 1040.
[XI A. L. Mackay, Artu Cr.wtaNogr. 1962. 15. 916.
[9] L. Fejes Tbth, Lugerringen in der Ebene uuf der Kugrl and im Raum,
Springer, Berlin, 1953, p. 171.
[lo] I. Stewart, New Scientist 1991, 130. No. 1777. 29.
[Ill N. M a x , Nature 1992, 355, 115.
[12] G. Higg, Ark Kern; Minerd. Geol. B 1943, 16, 1.
[13] G. S. Zhdanov. Dokl. Akud. Nuuk SSSR 1945. 4X. 40.
[14] P. A. Beck, Z . Krisldlogr. 1967. 124. 101
[15] T. J. McLarnan, Z. KristuNugr. 1981, 155, 269.
[16] G. Polya. Acta Murh. 1937, 68, 145.
[17] D. E. White, Dircre/e Marh. 1975, 13, 277.
[IS] T. J. McLarnan. Z. Krblullogr.. 1981. 1.55, 277.
(191 U . Miiller. Acra C r w M o g r . Set/. B 1992. 48, in press.
0570-0833/9210606-0728 $3.50+ .25jO
Angrw. Chrm. Inr. Ed. Engl. 3f (1992) No. 6
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