Contractions of Polygons in Abstract Polytopes by Ilya Scheidwasser B.S. in Computer Science, University of Massachusetts Amherst M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy March 31, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Acknowledgements First, I would like to thank my advisor, Professor Egon Schulte. From the first class I took with him in the first semester of my Masters program, Professor Schulte was an engaging, clear, and kind lecturer, deepening my appreciation for mathematics and always more than happy to provide feedback and answer extra questions about the material. Every class with him was a sincere pleasure, and his classes helped lead me to the study of abstract polytopes. As my advisor, Professor Schulte provided me with invaluable assistance in the creation of this thesis, as well as career advice. For all the time and effort Professor Schulte has put in to my endeavors, I am greatly appreciative. I would also like to thank my dissertation committee for taking time out of their sched- ules to provide me with feedback on this thesis. In addition, I would like to thank the various instructors I've had at Northeastern over the years for nurturing my knowledge of and interest in mathematics. I would like to thank my peers and classmates at Northeastern for their company and their assistance with my studies, and the math department's Teaching Committee for the privilege of lecturing classes these past several years. Finally, I would like to thank my girlfriend Rachel for her support and company during my work on this dissertation. Her presence in my life has been a priceless grounding force. She helps keep me happy and sane. ii Abstract of Dissertation There are several well-known constructions of new polytopes from old, such as the pyra- mid and prism constructions. This thesis defines two new local constructions on abstract polytopes. The first construction, called digonal contraction, allows digonal sections to be removed by merging their two edges into a single edge. The second construction, called polygonal contraction, allows polygonal sections with at least four vertices to be converted to two smaller polygons by merging two non-adjacent vertices. Neither of these contractions can be applied arbitrarily; we present necessary and sufficient conditions for the use of each contraction. In the case of polygonal contraction, we find that the graph-theoretical proper- ties of the polytope affect whether the contraction can be applied. Both of these contractions have interesting interactions with another new construction, the k-bubble construction in- troduced by Helfand (2013, Section 2.2), which are described. We also investigate how polygonal contraction can be performed somewhat globally, and what the effects are on the automorphism group of the polytope. iii Table of Contents Acknowledgements ii Abstract of Dissertation iii Table of Contents iv List of Figures vii 1 Introduction 1 2 Basic Notions 4 2.1 Basics of Convex Polytopes . .4 2.2 Posets and Symmetry . .6 2.3 Abstract Polytopes . 13 2.4 Graph Theory and Polytopes . 17 3 Digonal Contraction 21 3.1 Preliminary Results for Digonal Contraction . 21 3.2 The Digonal Contraction Theorem . 23 3.3 Reversing Digonal Contraction . 34 4 The Helfand Construction 38 4.1 The Global and Local Helfand Constructions . 38 4.2 Connections to Digonal Contraction . 43 5 Polygonal Contraction 51 5.1 Defining Polygonal Contraction . 51 iv 5.2 The Polygonal Contraction Theorem . 57 5.3 Polygonal Contraction for Polytope Lattices . 75 5.4 Connections to the Helfand Construction . 76 6 Polygonal Contraction and Symmetry 84 6.1 Multiple Polygonal Contraction . 84 6.2 Connections to Symmetry . 87 Bibliography 104 v List of Figures 2.1 A flag of a 3-cube. .6 2.2 The Hasse diagram of a square. .8 2.3 The pyramid and prism over a pentagon. 10 2.4 The distinguished generators of a square. 10 2.5 A triangular 2-section of a cube. 12 2.6 Examples of abstract polytopes which are not convex polytopes. 15 3.1 Digonal contraction on a cube-like polytope. 24 3.2 Four non-contractible digons. 31 3.3 An example of Condition (4) breaking for digonal contraction. 33 3.4 Merging two polygons with digonal contraction. 35 4.1 Two different applications of the global Helfand construction to the 3-cube. 40 4.2 Producing a digon via the Helfand construction. 44 4.3 Producing and contracting digons with the Helfand construction. 45 4.4 Merging two triangles into a square with digonal contraction. 48 4.5 Merging two polygons with digonal contraction, in connection with the Helfand construction. 50 5.1 Polygonal contraction on a hexagon. 52 5.2 A polytope nontrivially satisfying the conditions in Prop. 5.1.2. 55 5.3 Edge graphs satisfying and violating the conditions in Prop. 5.1.2. 56 5.4 Polygonal contraction failing due to apeirogons. 67 5.5 Examples of connectivity and non-connectivity in polygonal contraction. 68 5.6 Two similar polygons with different results under polygonal contraction. 70 vi 5.7 Five different examples of non-contractible polygonal sections F=G in poly- topes. 71 5.8 The edge graphs G for the examples in Figure 5.7. 72 5.9 Breaking or preserving the lattice condition via polygonal contraction. 77 5.10 Producing and contracting polygons with the Helfand construction. 80 5.11 The creation of a triangle via the Helfand construction. 82 5.12 The creation of another triangle via the Helfand construction. 82 6.1 The truncated square tiling. 91 6.2 The polyhedron P1 obtained by contracting all octagons in the truncated square tiling. 92 6.3 The polyhedron P2 obtained by contracting all squares in the truncated square tiling. 93 6.4 The polyhedron P12 obtained by contracting all squares and octagons all in the truncated square tiling. 95 6.5 Further contraction of the truncated square tiling. 96 6.6 The first step in a maximal contraction of the truncated square tiling. 97 6.7 The second step in a maximal contraction of the truncated square tiling. 98 6.8 The third step in a maximal contraction of the truncated square tiling. 99 6.9 The final step in a maximal contraction of the truncated square tiling. 100 vii Chapter 1 Introduction An abstract polytope is a combinatorial object generalizing the face lattice of a convex polytope (McMullen and Schulte, 2002, Section 2A). Specifically, an abstract polytope is a poset. The elements of the set represent faces of the polytope and one face being less than another in the poset represents two faces being incident, such as a vertex contained in an edge. In order for a poset to be an abstract polytope, it must satisfy four conditions - some are very straightforward, such as the requirement that there is exactly one least face and one greatest face, while others are more complicated, such as the strong connectivity condition. As such, testing whether an arbitrary poset is an abstract polytope is not simple. A construction on abstract polytopes is a way to perturb an abstract polytope, whether locally or globally, in order to get another abstract polytope. Two well-known constructions are the pyramid construction and the prism construction: just as we can take the pyramid over a convex polytope or the prism over a convex polytope to get a new convex polytope of one higher dimension (Gr¨unbaum, 2003, Sections 4.2, 4.4), we can generalize these con- structions to apply to abstract polytopes, and performing either of these constructions on an abstract polytope always produces an abstract polytope of one higher rank (dimension). Helfand introduced a new global construction on abstract polytopes called the k-bubble construction (Helfand, 2013), or in this paper, the global Helfand construction (we define a local version later). This construction is a generalization of the operation of truncating a convex polytope. For example, if we truncate a cube, we imagine slicing the cube near each of its vertices, so that there is now a new triangle for every vertex; the resulting convex polytope is the well-known truncated cube. This new polytope is not regular like the original polytope was, but is still highly symmetric. The global Helfand construction allows us to 1 apply a similar operation, in various ways, to an abstract polytope, and always produces a new abstract polytope of the same rank. The two new constructions examined in this thesis are local constructions focused on polygons in abstract polytope. A polygon is any two-dimensional section of the face poset of a polytope: for example, the cube clearly has square faces, but it also has triangular sections (its vertex figures). The first construction examined is the digonal contraction. A digon is an abstract polygon with just two vertices and two edges, and the presence of digonal sections in an abstract polytope is sometimes viewed as a degenerate condition. Digonal contraction is the operation of taking a digonal section, removing the greatest face of that section altogether, and merging the two edges into a single edge. The second construction examined is the polygonal contraction. Unlike digonal contraction, the aim here is not to remove the polygon altogether, but to split it into two smaller polygons. This is done by taking two non-adjacent vertices of the polygon (hence, the polygon must be at least a square) and merging them, thus producing two smaller polygons in a natural way. Neither of these contractions can be applied to arbitrary polygons in arbitrary polytopes; both have various conditions that must be satisfied in order for the contraction to produce an abstract polytope (otherwise, the contraction will just produce a poset which does not qualify as an abstract polytope for one reason or another).