On the Multiple Packing Densities of Triangles (original) (raw)
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On the Multiple Packing Densities of Triangles Kirati Sriamorn
2016
Given a convex disk K and a positive integer k, let δ T (K) and δ k L(K) denote the k-fold translative packing density and the k-fold lattice packing density of K, respectively. Let T be a triangle. In a very recent paper [2], I proved that δ L(T ) = 2k 2 2k+1 . In this paper, I will show that δ T (T ) = δ k L(T ).
On the Multiple Covering Densities of Triangles
Discrete & Computational Geometry, 2015
Given a convex disk K and a positive integer k, let ϑ k T (K) and ϑ k L (K) denote the k-fold translative covering density and the k-fold lattice covering density of K, respectively. Let T be a triangle. In a very recent paper, K. Sriamorn [4] proved that ϑ k L (T) = 2k+1 2. In this paper, we will show that ϑ k T (T) = ϑ k L (T).
On the Lattice Packings and Coverings of Convex
2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.
Packing and Covering with Centrally Symmetric Convex Disks
Discrete & Computational Geometry, 2014
Given a convex disk K (a convex compact planar set with nonempty interior), let δ L (K) and θ L (K) denote the lattice packing density and the lattice covering density of K, respectively. We prove that for every centrally-symmetric convex disk K we have that 1 ≤ δ L (K)θ L (K) ≤ 1.17225. .. The left inequality is tight and it improves a 10-year old result. Keywords Arrangements of convex disks • Packing density • Covering density 1 Introduction In this paper, we consider arrangements of convex disks in the Euclidean plane. A convex disk is a compact convex set with nonempty interior; its area will be denoted by A(K). An arrangement of congruent copies (translates) of a convex disk K is a family A of convex disks, each of which is congruent to (is a translate of) K. The arrangement is a packing if its members' interiors are mutually disjoint, and it is a covering if the union of its members is the whole plane.
On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals
arXiv (Cornell University), 2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.
The Set of Packing and Covering Densities of Convex Disks
Discrete & Computational Geometry, 2013
For every convex disk K (a convex compact subset of the plane, with non-void interior), the packing density δ(K) and covering density ϑ(K) form an ordered pair of real numbers, i.e., a point in R 2. The set Ω consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on δ(K) and ϑ(K) jointly outline a relatively small convex polygon P that contains Ω, while the exact shape of Ω remains a mystery. Here we describe explicitly a leaf-shaped convex region Λ contained in Ω and occupying a good portion of P. The sets ΩT and ΩL of translational packing and covering densities and lattice packing and covering densities are defined similarly, restricting the allowed arrangements of K to translated copies or lattice arrangements, respectively. Due to affine invariance of the translative and lattice density functions, the sets ΩT and ΩL are compact. Furthermore, the sets Ω, ΩT and ΩL contain the subsets Ω , Ω T and Ω L respectively, corresponding to the centrally symmetric convex disks K,
Dense packing of space with various convex solids
2010
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set K in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are several significant results on lattice packing in three dimensions as well. This article surveys the known results, focusing on the most recent progress. Also, many new problems are stated, indicating directions in which future development of the general packing theory in three dimensions seems feasible.
Optimal packing of similar triangles
Journal of Combinatorial Theory, Series A, 1995
We prove that any set of similar triangles with total area equal to a can be packed, using only translations and reflections, inside a similar triangle of area 2o~. The bound 2o~ cannot be improved in general. The proof is constructive, giving a fast algorithm for producing such a packing. We conjecture that such a set can be packed inside a similar triangle of area 2a using translations only. © 1995 Academic Press, Inc.
Discrete & Computational Geometry, 2001
Givenafamily C of planeconvexbodies,let n (C) bethesetof all pairs(x, y) with the propertythatthereexistsK E C suchthat iJ(K) = x and8(K) = y, whereiJ(K) and8(K) denotethedensitiesofthethinnestcoveringandthedensest packingoftheplane with copiesof K, respectively.The set nL(C) is definedanalogously, with the difference thatwe restrictour attentionto lattice packingsandcoverings. We provethat,for everycentrallysymmetricplaneconvexbody K, and give an exactanalytic descriptionof ndP s) whereP s is the family of all centrally symmetricoctagons. This allows us to showthatthe aboveinequalitiesareasymptotically tight.
Lower bounds for packing densities
Acta Mathematica Hungarica, 1991
If a packing of incompressible rigid convex objects is sufficiently compressed or "compacted", one expects that the packing density will not be small. The aim of this paper is to show that certain conditions on a packing insure that there is at least a lower bound on the packing density, which generalize some previous results concerning such lower bounds.