Gruber P. Convex and Discrete Geometry

30 Packing of Convex Bodies 439
The following is a list of the numbers of extreme forms for dimensions
d = 2,...,8:
d = 2: 1 : Korkin and Zolotarev [611]
d = 3: 1 : Korkin and Zolotarev [611]
d = 4: 2 : Korkin and Zolotarev [611]
d = 5: 3 : Korkin and Zolotarev [611]
d = 6: 6 : Hofreiter [518], Barnes [71, 72]
d = 7: 30 : Conway and Sloane [221]
d = 8: 2408 : Riener [836, 837]
For more information see Martinet [691].
Remark. Using Voronoĭ’s theorem, one can decide, at least in principle, whether
a given positive quadratic form is extreme or, equivalently, whether a given lattice
provides a locally densest packing of balls.
In order to find all extreme forms it is sufficient to consider a maximal set of
inequivalent vertices of R(m) and to take those vertices q for which the tangent hyperplane
of the discriminant surface through q meets R(m) only at q. Unfortunately,
no effective algorithm to determine the extreme forms is yet known.
30 Packing of Convex Bodies
The problem of packing of convex bodies and, in particular, of Euclidean balls has
attracted interest ever since Kepler first considered such questions. One reason for
this is that packing results can have interesting arithmetic interpretations. Among
the eminent contributors we mention Kelvin, Minkowski, Thue, Voronoĭ and Fejes
Tóth. The first investigations of non-lattice packings are due to Thue [996, 997], the
later development was strongly influenced by the seminal work of Fejes Tóth. As a
nice historical curiosity we cite the following observation of Reynolds [832] on the
distortion of a dense packing of grains of sand, see Coxeter [231]:
As the foot presses upon the sand when the falling tide leaves it firm, the portion of
it immediately surrounding the foot becomes momentarily dry. .... The pressureof
the foot causes dilatation of the sand, and so more water is [drawn] through the interstices
of the surrounding sand ...,leavingitdryuntilasufficientsupply has been
obtained from below, when it again becomes wet. On raising the foot we generally
see that the sand under and around it becomes wet for a little time. This is because
the sand contracts when the distorting forces are removed, and the excess of water
escapes at the surface.
At the beginning the sand grains form a dense packing. Then the pressure of the
foot distorts the packing which consequently becomes less dense and thus provides
more space for the water. By capillary forces the water then is drawn into the interior
of the sand. On raising the foot with the water as lubricant the sand grains again glide
back into a dense packing which provides less space for the water.