406 research outputs found
On compact packings of the plane with circles of three radii
A compact circle-packing of the Euclidean plane is a set of circles which
bound mutually disjoint open discs with the property that, for every circle
, there exists a maximal indexed set so that, for every , the circle is tangent to
both circles and
We show that there exist at most pairs with for
which there exist a compact circle-packing of the plane consisting of circles
with radii , and .
We discuss computing the exact values of such 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 ,
, allowing triangulated packings by discs of radii 1, and .
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 and . 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 . Meanwhile Gerd Blind in \cite{MR0275291,MR0377702} proved that for
, the most dense packing possible is , which is
when all the disks are the same size. In \cite{MR0165423}, the upper bound of
the ratio such that the density of his packings greater than
that Fejes T\'oth found was . Here we improve
that upper bound to . 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 and 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
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