65,509 research outputs found
The sphere packing problem in dimension 24
Building on Viazovska's recent solution of the sphere packing problem in
eight dimensions, we prove that the Leech lattice is the densest packing of
congruent spheres in twenty-four dimensions and that it is the unique optimal
periodic packing. In particular, we find an optimal auxiliary function for the
linear programming bounds, which is an analogue of Viazovska's function for the
eight-dimensional case.Comment: 17 page
New upper bounds on sphere packings I
We develop an analogue for sphere packing of the linear programming bounds
for error-correcting codes, and use it to prove upper bounds for the density of
sphere packings, which are the best bounds known at least for dimensions 4
through 36. We conjecture that our approach can be used to solve the sphere
packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur
On packing spheres into containers (about Kepler's finite sphere packing problem)
In an Euclidean -space, the container problem asks to pack equally
sized spheres into a minimal dilate of a fixed container. If the container is a
smooth convex body and we show that solutions to the container
problem can not have a ``simple structure'' for large . By this we in
particular find that there exist arbitrary small , such that packings in a
smooth, 3-dimensional convex body, with a maximum number of spheres of radius
, are necessarily not hexagonal close packings. This contradicts Kepler's
famous statement that the cubic or hexagonal close packing ``will be the
tightest possible, so that in no other arrangement more spheres could be packed
into the same container''.Comment: 13 pages, 2 figures; v2: major revision, extended result, simplified
and clarified proo
Spherical codes, maximal local packing density, and the golden ratio
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd
involves the placement of N nonoverlapping spheres of unit diameter near an
additional fixed unit-diameter sphere such that the greatest distance from the
center of the fixed sphere to the centers of any of the N surrounding spheres
is minimized. Solutions to the DLP problem are relevant to the realizability of
pair correlation functions for packings of nonoverlapping spheres and might
prove useful in improving upon the best known upper bounds on the maximum
packing fraction of sphere packings in dimensions greater than three. The
optimal spherical code problem in Rd involves the placement of the centers of N
nonoverlapping spheres of unit diameter onto the surface of a sphere of radius
R such that R is minimized. It is proved that in any dimension, all solutions
between unity and the golden ratio to the optimal spherical code problem for N
spheres are also solutions to the corresponding DLP problem. It follows that
for any packing of nonoverlapping spheres of unit diameter, a spherical region
of radius less than or equal to the golden ratio centered on an arbitrary
sphere center cannot enclose a number of sphere centers greater than one more
than the number that can be placed on the region's surface.Comment: 12 pages, 1 figure. Accepted for publication in the Journal of
Mathematical Physic
Mathematical optimization for packing problems
During the last few years several new results on packing problems were
obtained using a blend of tools from semidefinite optimization, polynomial
optimization, and harmonic analysis. We survey some of these results and the
techniques involved, concentrating on geometric packing problems such as the
sphere-packing problem or the problem of packing regular tetrahedra in R^3.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates
and correction
Random perfect lattices and the sphere packing problem
Motivated by the search for best lattice sphere packings in Euclidean spaces
of large dimensions we study randomly generated perfect lattices in moderately
large dimensions (up to d=19 included). Perfect lattices are relevant in the
solution of the problem of lattice sphere packing, because the best lattice
packing is a perfect lattice and because they can be generated easily by an
algorithm. Their number however grows super-exponentially with the dimension so
to get an idea of their properties we propose to study a randomized version of
the algorithm and to define a random ensemble with an effective temperature in
a way reminiscent of a Monte-Carlo simulation. We therefore study the
distribution of packing fractions and kissing numbers of these ensembles and
show how as the temperature is decreased the best know packers are easily
recovered. We find that, even at infinite temperature, the typical perfect
lattices are considerably denser than known families (like A_d and D_d) and we
propose two hypotheses between which we cannot distinguish in this paper: one
in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a
competitor, in which their packing fraction decreases super-exponentially,
namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also
find properties of the random walk which are suggestive of a glassy system
already for moderately small dimensions. We also analyze local structure of
network of perfect lattices conjecturing that this is a scale-free network in
all dimensions with constant scaling exponent 2.6+-0.1.Comment: 19 pages, 22 figure
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