Echoes of the hexagon: remnants of hexagonal packing inside regular polygons

Based on numerical simulations that we have carried out, we provide evidence that for regular polygons with $\sigma= 6j$ sides (with $j=2,3,\dots$), $N(k)=3 k (k+1)+1$ (with $k=1,2,\dots$) congruent disks of appropriate size can be nicely packed inside these polygons in highly symmetrical configurations which apparently have maximal density for $N$ sufficiently small. These configurations are invariant under rotations of $\pi/3$ and are closely related to the configurations with perfect hexagonal packing in the regular hexagon and to the configurations with {\sl curved hexagonal packing} (CHP) in the circle found long time ago by Graham and Lubachevsky. At the basis of our explorations are the algorithms that we have devised, which are very efficient in producing the CHP and more general configurations inside regular polygons. We have used these algorithms to generate a large number of CHP configurations for different regular polygons and numbers of disks; a careful study of these results has made possible to fully characterize the general properties of the CHP configurations and to devise a {\sl deterministic} algorithm that completely ensembles a given CHP configuration once an appropriate input ("DNA") is specified. Our analysis shows that the number of CHP configurations for a given $N$ is highly degenerate in the packing fraction and it can be explicitly calculated in terms of $k$ (number of shells), of the building block of the DNA itself and of the number of vertices in the fundamental domain (because of the symmetry we work in $1/6$ of the whole domain). With the help of our deterministic algorithm we are able to build {\sl all} the CHP configurations for a polygon with $k$ shells.Comment: 25 pages, 12 figures, 4 table

Circle packing in arbitrary domains

We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles, ellipses, crosses, multiply connected domains and even to the cardioid. For many of the cases that we have studied no previous result was available. The fundamental idea in our approach is the introduction of "image" disks, which allows one to work with a fixed container, thus lifting the limitations of the packing algorithms of \cite{Nurmela97,Amore21,Amore23}. We believe that the extension of our algorithm to three (or higher) dimensional containers (not considered here) can be done straightforwardly.Comment: 26 pages, 17 figure

Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.Comment: 18 page

Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{\Omega(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{\Omega(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

Collective Phenomena in Active Systems

This dissertation investigates collective phenomena in active systems of biological relavance across length scales, ranging from intracellular actin systems to bird flocks. The study has been conducted via theoretical modeling and computer simulations using tools from soft condensed matter physics and non-equilibrium statistical mechanics. The work has been organized into two parts through five chapters. In part one (chapter 2 to 3), continuum theories have been utilized to study pattern formation in bacteria suspensions, actomyosin systems and bird flocks, whose dynamics is described generically within the framework of polar active fluids. The continuum field equations have been written down phenomenogically and derived rigorously through explicit coarse-graining of corresponding microscopic equations of motion. We have investigated the effects of alignment interaction, active motility, non-conserved density, and rotational inertia on pattern formation in active systems. In part two (chapter 4 to 5), computer simulations have been performed to study the self-organization and mechanical properties of dense active systems. A minimal self-propelled particle (SPP) model has been utilized to understand the aggregation and segregation of active systems under confinement (Chapter 4), where an active pressure has been defined for the first time to characterize the mechanical state of the active system. The same model is utilized in Chapter 5 to understand the self-assembly of passive particles in an active bath