On compact packings of the plane with circles of three radii

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for which there exist a compact circle-packing of the plane consisting of circles with radii $s$, $r$ and $1$. We discuss computing the exact values of such $0<s<r<1$ as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.Comment: Dataset referred to in the text can be obtained at http://dx.doi.org/10.17632/t66sfkn5tn.

When Ternary Triangulated Disc Packings Are Densest: Examples, Counter-Examples and Techniques

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each "hole" is bounded by three pairwise tangent discs are called triangulated. Connelly conjectured that when such packings exist, one of them maximizes the proportion of the covered surface: this holds for unary and binary disc packings. For ternary packings, there are 164 pairs (r, s), 1 > r > s, allowing triangulated packings by discs of radii 1, r and s. In this paper, we enhance existing methods of dealing with maximal-density packings in order to study ternary triangulated packings. We prove that the conjecture holds for 31 triplets of disc radii and disprove it for 40 other triplets. Finally, we classify the remaining cases where our methods are not applicable. Our approach is based on the ideas present in the Hales\u27 proof of the Kepler conjecture. Notably, our proof features local density redistribution based on computer search and interval arithmetic

Density of triangulated ternary disc packings

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 15 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 40 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.Comment: 37 pages, SageMath code included in source (in 'code' directory

Maximally Dense Disc Packings on the Plane

Suppose one has a collection of disks of various sizes with disjoint interiors, a packing, in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book \textit{Regular Figures} \cite{MR0165423}, L\'aszl\'o Fejes T\'oth found a series of packings that were his best guess for the maximum density for any $1> q > 0.2$. Meanwhile Gerd Blind in \cite{MR0275291,MR0377702} proved that for $1\ge q > 0.72$, the most dense packing possible is $\pi/\sqrt{12}$, which is when all the disks are the same size. In \cite{MR0165423}, the upper bound of the ratio $q$ such that the density of his packings greater than $\pi/\sqrt{12}$ that Fejes T\'oth found was $0.6457072159..$. Here we improve that upper bound to $0.6585340820..$. Both bounds were obtained by perturbing a packing that has the property that the graph of the packing is a triangulation, which L. Fejes T\'oth called a \emph{compact} packing, and we call a \emph{triangulated} packing. Previously all of L. Fejes T\'oth's packings that had a density greater than $\pi/\sqrt{12}$ and $q > 0.35$ were based on perturbations of packings with just two sizes of disks, where the graphs of the packings were triangulations. Our new packings are based on a triangulated packing that have three distinct sizes of disks, found by Fernique, Hashemi, and Sizova, \cite{1808.10677}, which is something of a surprise. We also point out how the symmetries of a triangulated doubly periodic packing can by used to create the actual packing that is guaranteed by a famous result of Thurston, Andreev, and Andreeson \cite{MR2131318}.Comment: The main graph that shows the relation to previous packings has been changed and focused on the critical portion. Also various unneeded parts have been remove

Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores

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