Some progress in the packing of equal circles in a square

AbstractThe problem of the densest packing of n equal circles in a square has been solved for n<10 in [4, 6]; and some solutions have been proposed for n ⩾ 10. In this paper we give some better packings for n = 10, 11, 13 and 14

Packing 16, 17 of 18 circles in an equilateral triangle

We present new, efficient packings for 16, 17 and 18 congruent circles in an equilateral triangle. The results have been found by the use of simulated annealing and a quasi-Newton optimization technique, supplemented with some human intelligence

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 $\frac{n}{n+1}\frac{\pi}{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.Comment: 26 pages, 13 figure

Minimal surfaces from circle patterns: Geometry from combinatorics

We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal surfaces. The data used for the construction are purely combinatorial--the combinatorics of the curvature line pattern. A Weierstrass-type representation and an associated family are derived. We show the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended introduction. Minor changes in presentation. v3: revision according to the referee's suggestions, improved & expanded exposition, references added, minor mistakes correcte