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Pappus Configurations in Finite Hall Affine Planes PAPPUS CONFIGURATIONS IN FINITE HALL AFFINE PLANES by Lorinda A.M. Leshock A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Summer 2020 © 2020 Lorinda A.M. Leshock All Rights Reserved PAPPUS CONFIGURATIONS IN FINITE HALL AFFINE PLANES by Lorinda A.M. Leshock Approved: Louis Rossi, Ph.D. Chair of the Department of Mathematical Sciences Approved: John A. Pelesko, Ph.D. Dean of the College of Arts and Sciences Approved: Douglas J. Doren, Ph.D. Interim Vice Provost for Graduate and Professional Education and Dean of the Graduate College I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Felix Lazebnik, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Robert Coulter, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Jason Williford, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Qing Xiang, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS This thesis would not have existed without the guidance of Felix Lazebnik. I appreciate his ingenuity, fortitude, and graciousness as an advisor and a teacher. Learning to do mathematics with him has been an honor. I am thankful to Eric Moorhouse for the discussion of Hall planes, and in particular, for his help in understanding some properties of the collineation groups of Hall planes. Many thanks go to my committee members, Robert Coulter, Jason Williford, and Qing Xiang for their helpful comments and questions. I am grateful to the professors of the classes I took; their lectures and coursework nourished my mind. The staff of the Department of Mathematical Sciences was especially efficient and friendly. I feel fortunate to have met many graduate students who faced challenges with dignity and shared humor during our time at the University of Delaware. Finally, I want to acknowledge my family. Because of their love and support, I had the opportunity to begin and the stamina to complete this endeavor. They encouraged me every step along the way. iv TABLE OF CONTENTS LIST OF FIGURES .................................. vii ABSTRACT ...................................... viii Chapter 1 INTRODUCTION ................................ 1 1.1 Affine Planes . .2 1.2 Configurations, Classical Theorems, Coordinatization . .5 1.3 Structure of the Dissertation . .9 2 FINITE HALL PLANES ............................. 11 2.1 Analytic Construction . 11 2.1.1 Algebraic System . 11 2.1.2 Construction of Hall planes . 15 2.1.3 Symmetries . 17 2.1.4 Selected Properties . 25 2.1.5 Action of Collineation Group of AH on Pairs of Lines. 26 2.1.6 The Way Mathematica is Used . 28 3 MAIN RESULTS ................................. 30 3.1 Proof of Theorem 3.0.1........................... 34 3.1.1 Case 1: BF/BF ........................... 34 3.1.1.1 Intersecting `1, `2 ..................... 34 3.1.1.2 Parallel `1, `2 ....................... 41 3.1.2 Case 2: NBF/NBF ......................... 44 3.1.2.1 Intersecting `1, `2 ..................... 44 v 3.1.2.2 Parallel `1, `2 ....................... 54 3.2 Proof of Theorem 3.0.2........................... 61 3.2.1 Case 3a: NBF/BF .......................... 61 3.3 Proof of Theorem 3.0.3........................... 70 3.3.1 Case3b: BF/NBF .......................... 70 4 SUPPLEMENT .................................. 76 4.1 Numerical Search for Pappus Configurations . 76 4.2 Alternative Methods for Constructing Hall planes . 77 4.3 Possible Future Work . 79 5 CODE SAMPLES ................................. 80 5.1 Magma . 80 5.2 Mathematica . 81 BIBLIOGRAPHY ................................... 83 vi LIST OF FIGURES 1.1 Two diagrams of the plane on four points . .4 1.2 Two diagrams of the Desargues configuration . .7 1.3 Three diagrams of the Pappus configuration . .8 vii ABSTRACT In the classical projective planes, both Desargues’s theorem and Pappus’s theorem hold. According to a result of Ostrom, the Desargues configuration can also be found in every finite projective plane on at least twenty-one points, classical or not. In fact, Ostrom’s argument shows that the number of Desargues configurations in every finite plane is actually quite large. The result is also true in the finite projective plane on thirteen points. The existence of Pappus configurations in every non-classical finite affine or projective plane is unknown. We study whether the Pappus configuration is present in such planes. In particular, we endeavor to prove that in finite Hall affine planes, the fol- lowing strong version for the existence of Pappus configurations holds: For every pair of lines `1, `2 and every triple of points on `1 and every choice of a single point on `2, a pair of points on `2 can be found to complete a Pappus configuration. This statement is not proven in every case. When it is not, weaker versions for existence are shown. Hall planes are not Pappian, yet this work implies that the number of Pappus configurations in Hall planes is actually quite large. viii Chapter 1 INTRODUCTION Geometry was developed in many cultures to describe the world. In 300 BCE, Euclid’s Elements utilized an axiomatic deductive structure to organize mathematical knowledge, see Byer, Lazebnik, Smeltzer [3, p. 9]. By changing an axiom from Euclidean geometry in the 1820’s and 1830’s, two mathematicians, Lobachevsky and Bolyai independently developed non-Euclidean geometries. These mathematicians were contemporaries of Gauss who developed a geometry by changing the same axiom but chose not to publish it. Earlier speculations that such a geometry is possible are credited to Ibn al-Haytham (11th century), Khayyám (12th century), al-Tu¯ s¯ı (13th century), and Saccheri (18th century), see Rosenfeld and Yuschkevich [27]. Descartes formalized the connection between geometry and algebra by using coordinates and algebraic equations in the 17th century, see Brannan, Esplen, and Gray [2, p. 1]. Let us define two lines in the same plane to be parallel if they have no common point or they coincide. A problem determining whether two lines in the same plane are parallel may be solved geometrically, for example, by using alternate interior angles or analytically, by finding the equations of the lines and trying to solve the corresponding system of linear equations. This second approach to problem solving, the coordinate method, will be the one we favor in this dissertation. In 1899, Hilbert published the Foundations of Geometry. It is a succinct and groundbreaking text. In this book, Hilbert adjusted the axioms of Euclidean geometry with the goal of creating a simple and complete set of independent axioms from which a geometry can be built. As a part of Hilbert’s endeavor, his abstraction of the concepts of points, lines, and planes extended the scope of geometry. Furthermore, it promoted the use of the axiomatic method in mathematics. 1 1.1 Affine Planes Consider a set P, a point set, with elements called points and another set L, a line set, with elements called lines that are subsets of the point set. We say that a point P 2 P is on the line ` 2 L if P 2 `. In this situation, we may also say that the line ` is on the point P, that the line ` contains the point P, or that the point P and the line ` are incident. If a set of points is a subset of a line, then the points from the set are called collinear. If a point P is on every line from a set of lines, we say that the lines from the set are concurrent at P, or intersect at P. An affine plane is an ordered triple A = (P, L, I) where I ⊆ (P × L) [ (L × P) is a symmetric binary relation on P[L satisfying the following three axioms. A1. For every two distinct points, there exists a unique line containing them. A2. For every line ` and every point P not on `, there exists a unique line m on P such that m and ` are parallel. A3. There exist three noncollinear points. The concept of an affine plane reminds us of high school geometry. It is a generalization of the Euclidean plane with respect to only the notion of incidence. In the usual Euclidean affine plane we can introduce a coordinate system. Points are represented as ordered pairs of real numbers. Lines are described by linear equations with real coefficients. By using this approach, the Euclidean affine plane can be easily generalized to affine planes over other fields, in particular, over finite fields. The following interesting question was implicit in Hilbert’s Foundations and echoed by Artin [1, p. 51]: Given a plane geometry whose objects are the elements of two sets, the set of points and the set of lines; assume that certain axioms of geometric nature are true. Is it possible to find a field k such that the points of our geometry can be described by coordinates from k and the lines by linear equations? 2 For affine planes, this question was answered by Hilbert himself, and the answer was negative. Hilbert’s work inspired a search, not for a field k, but for a more general algebraic system fS, +, ·g, see Cerroni [5, p.
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