313 research outputs found
Highly saturated packings and reduced coverings
We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing with
congruent replicas of a body is -saturated if no members of it can
be replaced with replicas of , and it is completely saturated if it is
-saturated for each . Similarly, a covering with congruent
replicas of a body is -reduced if no members of it can be replaced
by replicas of without uncovering a portion of the space, and it is
completely reduced if it is -reduced for each . We prove that every
body in -dimensional Euclidean or hyperbolic space admits both an
-saturated packing and an -reduced covering with replicas of . Under
some assumptions on (somewhat weaker than convexity),
we prove the existence of completely saturated packings and completely reduced
coverings, but in general, the problem of existence of completely saturated
packings and completely reduced coverings remains unsolved. Also, we
investigate some problems related to the the densities of -saturated
packings and -reduced coverings. Among other things, we prove that there
exists an upper bound for the density of a -reduced covering of
with congruent balls, and we produce some density bounds for the
-saturated packings and -reduced coverings of the plane with congruent
circles
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
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