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The Contact Number Problem in Two and Three Dimensions University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2016 The Contact Number Problem in Two and Three Dimensions Foerster, Melanie Foerster, M. (2016). The Contact Number Problem in Two and Three Dimensions (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/28036 http://hdl.handle.net/11023/3302 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY The Contact Number Problem in Two and Three Dimensions by Melanie Foerster A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS CALGARY, ALBERTA September, 2016 c Melanie Foerster 2016 Abstract Given a packing of balls in two or three dimensions we want to maximize the number of contacts in the packing. This is referred to as the contact number problem. A packing of balls is totally separable if every two balls can be separated by a hyperplane that is disjoint from the interior of each ball. In this thesis we study the contact number problem for packings of congruent and incongruent balls in two and three dimensions, and we consider the totally separable version of each problem. ii Acknowledgements This thesis would not have been possible without the support of many important individuals. Most notably, I would like to extend sincere thanks to Dr. K´arolyBezdek. You have been an incredible, and invaluable, support both personally and academically over the last two years. It has been an honour to work with you. I will be forever grateful for your generosity. Thank you to my friends from coast to coast, especially my officemates Muhammad Khan, Michael Oliwa, and Trevor Song. This journey was made more successful with your help and encouragement. Special thanks to Dr. Robert Dawson and Dr. Dorette Pronk for igniting my love of research while I was studying in Halifax. Your support, and our stimulating mathematical conversations, have encouraged me throughout my studies. iii Table of Contents Abstract ii Acknowledgements iii Table of Contents iv List of Figures v List of Key Symbols vi 1 Introduction 1 1.1 Kissing Number . 1 1.2 Contact Number . 2 1.3 Motivation from Material Science . 3 1.4 Notation . 5 2 The Contact Number Problem in Two Dimensions 6 2.1 The Contact Number Problem for Congruent Disks . 6 2.1.1 Harborth's Theorem . 6 2.1.2 Harborth's Theorem for Totally Separable Packings . 10 2.1.3 Digital Packings . 15 2.2 The Contact Number Problem for Incongruent Disks . 16 2.2.1 Koebe-Andreev-Thurston Representation Theorem . 16 2.2.2 Totally Separable Incongruent Packings . 23 3 The Contact Number Problem in Three Dimensions 28 3.1 The Contact Number Problem for Congruent Balls . 28 3.1.1 Voronoi Diagrams . 28 3.1.2 The Isoperimetric Inequality . 29 3.1.3 Congruent Ball Packings . 32 3.1.4 Totally Separable Congruent Ball Packings . 42 3.2 The Contact Number Problem for Incongruent Balls . 49 3.2.1 Kuperberg-Schramm Theorem . 50 3.2.2 Totally Separable Incongruent Ball Packings . 57 Bibliography 58 iv List of Figures 1.1 Kissing number . 2 1.2 Contact graph . 3 1.3 Totally separable packing . 4 2.1 Contact graph . 7 2.2 Contact graph after polygon deletion . 9 2.3 Congruent packing . 11 2.4 Contact graph of a totally separable packing . 12 2.5 Contact graph of a totally separable packing after polygon deletion . 13 2.6 Totally separable congruent digital packing . 15 2.7 Incongruent packing . 16 2.8 Wooden triangles . 18 2.9 Simplex in R3 ................................... 19 2.10 Wooden triangle vertex converging to π ..................... 20 2.11 Decomposition of a polygon by lines . 24 2.12 Decomposition of a polygon into convex polygons . 24 2.13 Totally separable incongruent packing in a triangle . 25 2.14 Totally separable incongruent packing . 26 2.15 Totally separable incongruent packing of 7 disks . 26 2.16 Totally separable incongruent packing of 10 disks . 27 3.1 Voronoi diagrams . 29 3.2 Triangle 4o1pq ................................. 33 3.3 Plane reflections . 34 3.4 Voronoi cells for a packing . 36 3.5 Surface volume contribution . 40 3.6 Voronoi cells for a totally separable packing . 45 3.7 Overlapping expanded balls . 48 3.8 Intersection of S(B) and S(C).......................... 52 3.9 Definition of width . 53 3.10 Width in terms of θ and ρr(B) ......................... 53 v List of Key Symbols Symbol or abbreviation Definition bxc Lower integer of x jE(G)j Number of edges in the graph G GnH Graph H deleted from the graph G bd(x) Boundary of x vol3(x) 3-dimensional volume of x svol2(x) 2-dimensional surface volume of x vi Chapter 1 Introduction Discrete geometry is the study of finite sets of points, lines, balls, convex sets, and convex polytopes. In discrete geometry, questions are focused on combinatorial properties. Such questions include: How many regions can a set of lines partition the plane into? How many times does the minimum distance occur in a set of points in the plane? Some problems date back hundreds of years, and many problems are motivated by other areas of mathematics and science; techniques used in discrete geometry are thus of great use to other disciplines [24]. 1.1 Kissing Number In order to discuss the contact number problem we first must describe the kissing number problem. The kissing number, denoted k(d), is the number of non-overlapping unit balls (of unit radius) that can touch a given unit ball in the d-dimensional Euclidean space Ed. The kissing number problem asks for the largest number of unit balls that can touch a given unit ball without overlap. The problem arose from a 1694 disagreement between Newton and Gregory regarding the number of 3-dimensional unit balls that could touch a central unit ball without overlap [8]. Newton thought the answer was 12, whereas Gregory believed the answer was 13. The question was answered many years later when Newton was proved 1 correct [34]. Figure 1.1: The kissing number in two dimensions, k(2) = 6. The known values of k(d) are k(2) = 6 (trivial, see Figure 1.1), k(3) = 12 ([34]), k(4) = 24 ([27]), k(8) = 240 ([28]), and k(24) = 196560 ([28]). The kissing number problem is closely related to the more general problem of finding bounds for optimal ball packings. 1.2 Contact Number A finite packing of balls in Ed is a finite family of non-overlapping balls in Ed. In this thesis, we only consider finite packings, so we will use \packing" to refer to a “finite packing". The contact graph of a packing of balls in Ed is a graph whose vertices correspond to the packing elements and whose two vertices are adjacent if and only if the corresponding packing elements touch each other. Figure 1.2 is an example of a packing with its corresponding contact graph. The number of edges in a contact graph is the contact number of the packing. The contact number problem asks for the largest possible contact number c(n) of a contact graph. That is, the maximum number of edges that a contact graph of n non-overlapping balls can have in Ed. The contact number problem is equivalent to Erd}os'srepeated shortest distance problem, which asks for the largest number of repeated shortest distances among n points in Ed. The planar case was first raised by Erd}osin 1946 [11]. Reutter conjectured an answer in 1972, 2 Figure 1.2: A packing with its corresponding contact graph. which was proved by Harborth in 1974 [16]. The more general forms of the problem were popularized by Erd}osand Ulam. As a close combinatorial relative, it is natural to investigate the maximum contact number of totally separable packings of balls in Ed. The notion of totally separable packings is an extension of separable domains, first introduced by Erd}os.Total separability was introduced in [12] by G. Fejes T´othand L. Fejes T´oth. A packing of balls in Ed is totally separable if every two balls can be separated by a hyperplane in Ed such that it is disjoint from the interior of each ball in the packing. An example of a totally separable packing in the plane is shown in Figure 1.3. The two dimensional version of the contact number problem for totally separable packings will be explored in Sections 2.1.2 and 2.2.2, and in three dimensions in Sections 3.1.4 and 3.2.2. 1.3 Motivation from Material Science As previously mentioned, discrete geometric techniques are of great use in other disciplines. In particular, the contact number problem has many important applications. Consider balls that are impenetrable and have short-range attractive forces. Packings of these balls are excellent models for materials such as powders, gels, and glasses [18]. The particles in these 3 Figure 1.3: A totally separable packing. materials are like hard balls that self-assemble due to their attractive forces. This process of self-assembly is of great interest to material scientists, chemists, statistical physicists, and biologists.
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