Aste T., Weaire D. Pursuit of perfect packing (IOP 2000)(147s).pdf

32 Hard problems with hard spheres
the lower. In the former mode any pellet is touched by four neighbours
in the same plane, and by one above and one below, and so on throughout,
each touched by six others. The arrangement will be cubic, and the
pellets, when subjected to pressure, will become cubes. But this will
not be the tightest packing. In the second mode not only is every pellet
touched by it four neighbours in the same plane, but also by four in the
plane above and by four below, and so throughout one will be touched
by twelve, and under pressure spherical pellets will become rhomboid.
This arrangement will be more comparable to the octahedron and pyramid.
This arrangement will be the tightest possible, so that in no other
arrangement could more pellets be packed into the same container 15 .
The structure here described by Kepler is cubic close packing, also called
face-centred cubic (fcc). It has the greengrocer’s packing fraction ¼¼ .
It is a regular structure: the local configurations repeat periodically in space like
wall paper but in three dimensions. Such a periodic structure is now often called
crystalline because it corresponds to the internal structure of crystals.
Kepler’s work was the first attempt to associate the external geometrical
shape of crystals with their internal composition of regularly packed microscopic
elements. It was very unusual for his time, when the word ‘crystal’ was applied
only to quartz, which was thought to be permanently frozen ice.
Kepler asserted that the cubic close packing ‘will be the tightest possible,
so that in no other arrangements could more pellets be packed into the same
container’. Despite Kepler’s confidence this conjecture long resisted proof and
became the oldest unsolved problem in discrete geometry.
3.9 Marvellous clarity: the life of Kepler
Kepler was born in Weil der Stadt (near Leonenberg, Germany) in 1571. He was
originally destined for the priesthood, but instead took up a position as school
teacher of mathematics and astronomy in Graz.
When Kepler arrived in Graz he was 25 years old and much occupied with
astrology. He issued a calendar and prognostication for 1595 which contained
predictions of bitter cold, a peasant uprising and invasions by the Turks. All were
fulfilled, greatly enhancing his local reputation.
Kepler was an enthusiastic Copernican. Today he is chiefly remembered for
his three laws on planetary motion but his search for cosmic harmonies was much
broader, ranging from celestial physics to sphere packings.
Kepler’s personality has been described as ‘neurotically anxious’. Certainly
he had an unhappy personal life. The story goes that, in seeking to optimize the
partner for his second marriage, he carefully analysed the merits of no less than
½ Quoted from: Kepler 1966 The Six-cornered Snowflake translated by C Hardie (Oxford: Clarendon).