773 research outputs found
Periodic Planar Disk Packings
Several conditions are given when a packing of equal disks in a torus is
locally maximally dense, where the torus is defined as the quotient of the
plane by a two-dimensional lattice. Conjectures are presented that claim that
the density of any strictly jammed packings, whose graph does not consist of
all triangles and the torus lattice is the standard triangular lattice, is at
most , where is the number of packing
disks. Several classes of collectively jammed packings are presented where the
conjecture holds.Comment: 26 pages, 13 figure
A combinatorial Yamabe flow in three dimensions
A combinatorial version of Yamabe flow is presented based on Euclidean
triangulations coming from sphere packings. The evolution of curvature is then
derived and shown to satisfy a heat equation. The Laplacian in the heat
equation is shown to be a geometric analogue of the Laplacian of Riemannian
geometry, although the maximum principle need not hold. It is then shown that
if the flow is nonsingular, the flow converges to a constant curvature metric.Comment: 20 pages, 5 figures. The paper arxiv:math.MG/0211195 was absorbed
into its new version and this pape
Notions of denseness
The notion of a completely saturated packing [Fejes Toth, Kuperberg and
Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125
(1998) 127-145] is a sharper version of maximum density, and the analogous
notion of a completely reduced covering is a sharper version of minimum
density. We define two related notions: uniformly recurrent and weakly
recurrent dense packings, and diffusively dominant packings. Every compact
domain in Euclidean space has a uniformly recurrent dense packing. If the
domain self-nests, such a packing is limit-equivalent to a completely saturated
one. Diffusive dominance is yet sharper than complete saturation and leads to a
better understanding of n-saturation.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.htm
Thurston's sphere packings on 3-dimensional manifolds, I
Thurston's sphere packing on a 3-dimensional manifold is a generalization of
Thusrton's circle packing on a surface, the rigidity of which has been open for
many years. In this paper, we prove that Thurston's Euclidean sphere packing is
locally determined by combinatorial scalar curvature up to scaling, which
generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere
packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity
that Thurston's Euclidean sphere packing can not be deformed (except by
scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife
Electrical networks and Stephenson's conjecture
In this paper, we consider a planar annulus, i.e., a bounded, two-connected,
Jordan domain, endowed with a sequence of triangulations exhausting it. We then
construct a corresponding sequence of maps which converge uniformly on compact
subsets of the domain, to a conformal homeomorphism onto the interior of a
Euclidean annulus bounded by two concentric circles. As an application, we will
affirm a conjecture raised by Ken Stephenson in the 90's which predicts that
the Riemann mapping can be approximated by a sequence of electrical networks.Comment: Comments are welcome
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