Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008)

Оценка числа решетчатых разбиений плоскости на центрально-симметричные полимино заданной площади

We study a problem about the number of lattice plane tilings by the given area centrosymmetrical polyominoes. A polyomino is a connected plane geomatric figure formed by joiining a finite number of unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a plane by polyominoes. In particular, the tiling is a lattice tiling if each tile can be mapped to any other tile by a translation which maps the whole tiling to itself. Earlier we proved that, for the number T(n) of a lattice plane tilings by polyominoes of an area n, holds the inequalities 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2, 7)n+1 . In the present work we prove a similar estimate for the number of lattice tilings with an additional central symmetry. Let Tc(n) be a number of lattice plane tilings by a given area centrosymmetrical polyominoes such that its translation lattice is a sublattice of Z 2 . It is proved that C1( √ 2)n ≤ Tc(n) ≤ C2n 2 ( √ 2.68)n . In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyominoes, and on the theory of self-avoiding walks on a square lattice.В работе рассматривается задача о числе решетчатых разбиений плоскости на центрально–симметричные полимино заданной площади. Полимино представляет собой связную фигуру на плоскости, составленную из конечного числа единичных квадратов, примыкающих друг к другу по сторонам. В настоящее время активно исследуются различные перечислительные комбинаторные задачи, связанные с полимино. Представляет интерес подсчет числа полимино определенных классов, а также подсчет числа разбиений конечных фигур или плоскости на полимино определенного типа. В частности, разбиение называется решетчатым, если любую фигуру разбиения можно перевести в любую другую фигуру параллельным переносом, переводящим все разбиение в себя. Ранее нами было доказано, что если T(n) – число решетчатых разбиений плоскости на полимино площади n, то справедливы неравенства 2 n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2, 7)n+1 . В настоящей работе мы получаем аналогичную оценку для числа решетчатых разбиений, дополнительно обладающих центральной симметрией. Пусть Tс(n) – число решетчатых разбиений плоскости на центрально–симметричные полимино площади n, решетка периодов которых является подрешеткой решетки Z 2 . В работе доказано, что C1( √ 2)n ≤ Tс(n) ≤ C2n 2 ( √ 2.68)n . При доказательстве нижней оценки исполь- зована явная конструкция, позволяющая построить требуемое число решетчатых разбиений плоскости. Доказательство верхней оценки основано на критерии существования решетчатого разбиения плоскости на полимино, а также на теории самонепересекающихся блужданий на квадратной решетке

Оценка числа решетчатых разбиений плоскости на полимино заданной площади

We study a problem of a number of lattice plane tilings by given area polyominoes. A polyomino is a connected plane geometric figure formed by joining edge to edge a finite number of unit squares. A tiling is a lattice tiling if each tile can be mapped to any other tile by translation which maps the whole tiling to itself. Let T(n) be a number of lattice plane tilings by given area polyominoes such that its translation lattice is a sublattice of Z². It is proved that 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyomino and on the theory of self-avoiding walk. Also, it is proved that almost all polyominoes that give lattice plane tilings have sufficiently large perimeters.Рассматривается задача о числе решетчатых разбиений плоскости на полимино заданной площади. Полимино представляет собой связную фигуру на плоскости, составленную из конечного числа единичных квадратов, примыкающих друг к другу по сторонам. Разбиение называется решетчатым, если любую фигуру разбиения можно перевести в любую другую фигуру параллельным переносом, переводящим все разбиение в себя. Пусть T(n) – число решетчатых разбиений плоскости на полимино площади n, решетка периодов которых является подрешеткой решетки Z² . Доказано, что 2 n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2.7)n+1. При доказательстве нижней оценки использована явная конструкция, позволяющая построить требуемое число решетчатых разбиений плоскости. Доказательство верхней оценки основано на одном критерии существования решетчатого разбиения плоскости на полимино, а также на теории самонепересекающихся блужданий на квадратной решетке. Также доказано, что почти все полимино, дающие решетчатые разбиения плоскости, имеют большой периметр

Snakes in the Plane

Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of polygons. I present a brief background of tiling theory and past work, with particular emphasis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new ``boundary-based'' technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite language of directions, to replace the ``area-based'' approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge assemblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier's recursive generation algorithm, and enumerates polysnakes using a novel approach. The shapes produced by the enumeration are subjected to tests to either determine their isohedral number or prove they are non-tiling. My results include the description of this novel approach to testing tiling properties, a correction to previous descriptions of the criteria for characterizing isohedral tilings, the verification of some previous results on polyforms, and the discovery of two new 4-anisohedral polysnakes

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection

Development of Atomistic Potentials for Silicate Materials and Coarse-Grained Simulation of Self-Assembly at Surfaces

This thesis is composed of two parts. The first is a study of evolutionary strategies for parametrization of empirical potentials, and their application in development of a charge-transfer potential for silica. An evolutionary strategy was meta-optimized for use in empirical potential parametrization, and a new charge-transfer empirical model was developed for use with isobaric-isothermal ensemble molecular dynamics simulations. The second is a study of thermodynamics and self-assembly in a particular class of athermal two-dimensional lattice models. The effects of shape on self-assembly and thermodynamics for polyominoes and tetrominoes were examined. Many interesting results were observed, including complex clustering, non-ideal mixing, and phase transitions. In both parts, computational efficiency and performance were important goals, and this was reflected in method and program development

Essays on Integer Programming in Military and Power Management Applications

This dissertation presents three essays on important problems motivated by military and power management applications. The array antenna design problem deals with optimal arrangements of substructures called subarrays. The considered class of the stochastic assignment problem addresses uncertainty of assignment weights over time. The well-studied deterministic counterpart of the problem has many applications including some classes of the weapon-target assignment. The speed scaling problem is of minimizing energy consumption of parallel processors in a data warehouse environment. We study each problem to discover its underlying structure and formulate tailored mathematical models. Exact, approximate, and heuristic solution approaches employing advanced optimization techniques are proposed. They are validated through simulations and their superiority is demonstrated through extensive computational experiments. Novelty of the developed methods and their methodological contribution to the field of Operations Research is discussed through out the dissertation

The Skyrme Model: Curved Space, Symmetries and Mass

The presented thesis contains research on topological solitons in (2+1) and (3+1) dimensional classical field theories, focusing upon the Skyrme model. Due to the highly non-linear nature of this model, we must consider various numerical methods to find solutions. We initially consider the (2+1) baby Skyrme model, demonstrating that the currently accepted form of minimal energy solutions, namely straight chains of alternating phase solitons, does not hold for higher charge. Ring solutions with relative phases changing by pi for even configurations or pi-pi/B for odd numbered configurations, are demonstrated to have lower energy than the traditional chain configurations above a certain charge threshold, which is dependant on the parameters of the model. Crystal chunk solutions are then demonstrated to take a lower energy but for extremely high values of charge. We also demonstrate the infinite charge limit of each of the above configurations. Finally, a further possibility of finding lower energy solutions is discussed in the form of soliton networks involving rings/chains and junctions. The dynamics of some of these higher charge solutions are also considered. In chapter 3 we numerically simulate the formation of (2+1)-dimensional baby Skyrmions from domain wall collisions. It is demonstrated that Skyrmion, anti-Skyrmion pairs can be produced from the interaction of two domain walls, however the process can require quite precise conditions. An alternative, more stable, formation process is proposed and simulated as the interaction of more than two segments of domain wall. Finally domain wall networks are considered, demonstrating how Skyrmions may be produced in a complex dynamical system. The broken planar Skyrme model, presented in chapter 4, is a theory that breaks global O(3) symmetry to the dihedral group D_N. This gives a single soliton solution formed of N constituent parts, named partons, that are topologically confined. We show that the configuration of the local energy solutions take the form of polyform structures (planar figures formed by regular N-gons joined along their edges, of which polyiamonds are the N=3 subset). Furthermore, we numerically simulate the dynamics of this model. We then consider the (3+1) SU(2) Skyrme model, introducing the familiar concepts of the model in chapter 5 and then numerically simulating their formation from domain walls. In analogue with the planar case, it is demonstrated that the process can require quite precise conditions and an alternative, more stable, formation process can be achieved with more domain walls, requiring far less constraints on the initial conditions used. The results in chapter 7 discuss the extension of the broken baby Skyrme model to the 3-dimensional SU(2) case. We first consider the affect of breaking the isospin symmetry by altering the tree level mass of one of the pion fields breaking the SO(3) isospin symmetry to an SO(2) symmetry. This serves to exemplify the constituent make up of the Skyrme model from ring like solutions. These rings then link together to form higher charge solutions. Finally the mass term is altered to allow all the fields to have an equivalent tree level mass, but the symmetry of the Lagrangian to be broken, firstly to a dihedral symmetry D_N and then to some polyhedral symmetries. We now move on to discussing both the baby and full SU(2) Skyrme models in curved spaces. In chapter 8 we investigate SU(2) Skyrmions in hyperbolic space. We first demonstrate the link between increasing curvature and the accuracy of the rational map approximation to the minimal energy static solutions. We investigate the link between Skyrmions with massive pions in Euclidean space and the massless case in hyperbolic space, by relating curvature to the pion mass. Crystal chunks are found to be the minimal energy solution for increased curvature as well as increased mass of the model. The dynamics of the hyperbolic model are also simulated, with the similarities and differences to the Euclidean model noted. One of the difficulties of studying the full Skyrme model in (3+1) dimensions is a possible crystal lattice. We hence reduce the dimension of the model and first consider crystal lattices in (2+1)-dimensions. In chapter 9 we first show that the minimal energy solutions take the same form as those from the flat space model. We then present a method of tessellating the Poincare disc model of hyperbolic space with a fundamental cell. The affect this may have on a resulting Skyrme crystal is then discussed and likely problems in simulating this process. We then consider the affects of a pure AdS background on the Skyrme model, starting with the massless baby Skyrme model in chapter 10. The asymptotics and scale of charge 1 massless radial solutions are demonstrated to take a similar form to those of the massive flat space model, with the AdS curvature playing a similar role to the flat space pion mass. Higher charge solutions are then demonstrated to exhibit a concentric ring-like structure, along with transitions (dubbed popcorn transitions in analogy with models of holographic QCD) between different numbers of layers. The 1st popcorn transitions from an n layer to an n+1-layer configuration are observed at topological charges 9 and 27 and further popcorn transitions for higher charges are predicted. Finally, a point-particle approximation for the model is derived and used to successfully predict the ring structures and popcorn transitions for higher charge solitons. The final chapter considers extending the results from the penultimate chapter to the full SU(2) model in a pure AdS_4 background. We make the prediction that the multi-layered concentric ring solutions for the 2-dimensional case would correlate a multi-layered concentric rational map configuration for the 3-dimensional model. The rational map approximation is extended to consider multi-layered maps and the energies demonstrated to reduce the minimal energy solution for charge B=11 which is again dubbed a popcorn transition. Finally we demonstrate that the multi shell structure extends to the full field solutions which are found numerically. We also discuss the affect of combined symmetries on the results which (while likely to be important) appear to be secondary to the dominant effective potential of the metric which simulates a packing problem and hence forces the popcorn transitions to act accordingly with the 2-dimensional model