8,315 research outputs found
Average kissing numbers for non-congruent sphere packings
The Koebe circle packing theorem states that every finite planar graph can be
realized as the nerve of a packing of (non-congruent) circles in R^3. We
investigate the average kissing number of finite packings of non-congruent
spheres in R^3 as a first restriction on the possible nerves of such packings.
We show that the supremum k of the average kissing number for all packings
satisfies
12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928
We obtain the upper bound by a resource exhaustion argument and the upper
bound by a construction involving packings of spherical caps in S^3. Our result
contradicts two naive conjectures about the average kissing number: That it is
unbounded, or that it is supremized by an infinite packing of congruent
spheres.Comment: 6 page
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
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