7 research outputs found
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
Density bounds for outer parallel domains of unit ball packings
We give upper bounds for the density of unit ball packings relative to their
outer parallel domains and discuss their connection to contact numbers. Also,
packings of soft balls are introduced and upper bounds are given for the
fraction of space covered by them.Comment: 22 pages, 1 figur
On the Minimum Ropelength of Knots and Links
The ropelength of a knot is the quotient of its length and its thickness, the
radius of the largest embedded normal tube around the knot. We prove existence
and regularity for ropelength minimizers in any knot or link type; these are
curves, but need not be smoother. We improve the lower bound for the
ropelength of a nontrivial knot, and establish new ropelength bounds for small
knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and
corrections; new section on asymptotic growth of ropelength; several new
reference
The Plane-Width of Graphs
Map vertices of a graph to (not necessarily distinct) points of the plane so
that two adjacent vertices are mapped at least a unit distance apart. The
plane-width of a graph is the minimum diameter of the image of the vertex set
over all such mappings. We establish a relation between the plane-width of a
graph and its chromatic number, and connect it to other well-known areas,
including the circular chromatic number and the problem of packing unit discs
in the plane. We also investigate how plane-width behaves under various
operations, such as homomorphism, disjoint union, complement, and the Cartesian
product