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Hyperbolic Geometry, Elliptic Geometry, and Euclidean Geometry

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Hyperbolic Geometry, Elliptic Geometry, and Euclidean Geometry Geometry with an Introduction to Cosmic Topology 2018 Edition Michael P. Hitchman Michael P. Hitchman Department of Mathematics Linfield College McMinnville, OR 97128 http://mphitchman.com © 2017-2020 by Michael P. Hitchman This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons. org/licenses/by-sa/4.0/. 2018 Edition Compiled October, 2020 ISBN-13: 978-1717134813 ISBN-10: 1717134815 A current version can always be found for free at http://mphitchman.com Cover image: Five geodesic paths between two points in a two-holed torus with constant curvature. Preface Geometry with an Introduction to Cosmic Topology approaches geometry through the lens of questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have an edge? Is it infinitely big? This text develops non-Euclidean geometry and geometry on surfaces at a level appropriate for undergraduate students who have completed a multivariable calculus course and are ready for a course in which to practice the habits of thought needed in advanced courses of the undergraduate mathematics curriculum. The text is also suited to independent study, with essays and discussions throughout. Mathematicians and cosmologists have expended considerable amounts of effort investigating the shape of the universe, and this field of research is called cosmic topology. Geometry plays a fundamental role in this research. Under basic assumptions about the nature of space, there is a simple relationship between the geometry of the universe and its shape, and there are just three possibilities for the type of geometry: hyperbolic geometry, elliptic geometry, and Euclidean geometry. These are the geometries we study in this text. Chapters 2 through 7 contain the core mathematical content. The text follows the Erlangen Program, which develops geometry in terms of a space and a group of transformations of that space. Chapter 2 focuses on the complex plane, the space on which we build two-dimensional geometry. Chapter 3 details transformations of the plane, including Möbius transformations. This chapter marks the heart of the text, and the inversions in Section 3.2 mark the heart of the chapter. All non-Euclidean transformations in the text are built from inversions. We formally define geometry in Chapter 4, and pursue hyperbolic and elliptc geometry in Chapters 5 and 6, respectively. Chapter 7 begins by extending these geometries to different curvature scales. Section 7.4 presents a unified family of geometries on all curvature scales, emphasizing key results common to them all. Section 7.5 provides an informal development of the topology of surfaces, and Section 7.6 relates the topology of surfaces to geometry, culminating with the Gauss-Bonnet formula. Section 7.7 discusses quotient spaces, and presents an important tool of cosmic topology, the Dirichlet domain. Two longer essays bookend the core content. Chapter 1 introduces the geo- metric perspective taken in this text. In my experience it is very helpful to spend time discussing this content in class. The Coneland and Saddleland activities iii iv (Example 1.3.5 and Example 1.3.7) have proven particularly helpful for motivating the content of the text. In Chapter 8, after having developed two-dimensional non-Euclidean geometry and the topology of surfaces, we glance meaningfully at the present state of research in cosmic topology. Section 8.1 offers a brief survey of three-dimensional geometry and 3-manifolds, which provide possible shapes of the universe. Sections 8.2 and 8.3 present two research programs in cosmic topology: cosmic crystallography and circles-in-the-sky. Measurements taken and analyzed over the last twenty years have greatly altered the way many cosmologists view the universe, and the text ends with a discussion of our present understanding of the state of the universe. Compass and ruler constructions play a visible role in the text, primarily because inversions are emphasized as the basic building blocks of transformations. Constructions are used in some proofs (such as the Fundamental Theorem of Möbius Transformations) and as a guide to definitions (such as the arc-length dif- ferential in the hyperbolic plane). We encourage readers to practice constructions as they read along, either with compass and ruler on paper, or with software such as The Geometer’s Sketchpad or Geogebra. Some Geometer’s Sketchpad templates and activites related to the text can be found at the text’s website. Reading the text online. An online text is fabulous at linking content, but we emphasize that this text is meant to be read. It was written to tell a mathematical story. It is not meant to be a collection of theorems and examples to be consulted as a reference. As such, online readers of this text are encouraged to turn the pages using the “arrow” buttons on the page as opposed to clicking on section links. Read the content slowly, participate in the examples, and work on the exercises. Grapple with the ideas, and ask questions. Feel free to email the author with questions or comments about the material. Changes from the previously published version. For those familiar with the oringial version of the text published by Jones & Bartlett, we note a few changes in the current edition. First, the numbering scheme has changed, so Example and Theorem and Figure numbers will not match the old hard copy. Of course the numbering schemes on the website and the new print options of the text do agree. Second, several exercises have been added. In sections with additional exercises, the new ones typically appear at the end of the section. Finally, Chapter 7 has been reorganized in an effort to place more emphasis on the family (Xk,Gk), and the key theorems common to all these geometries. This family now receives its own section, Section 7.4. The previous Section 7.4 (Observing Curvature in a Universe) has been folded into Section 7.3. Finally, the essays in Chapter 8 on cosmic topology and our understanding of the universe have been updated to include research done since the original publication of this text, some of which is due to sharper measurements of the temperature of the cosmic microwave background radiation obtained with the launch of the Planck satellite in 2009. Acknowledgements Many people helped with the development of this text. Jeff Weeks’ text The Shape of Space inspired me to first teach a geometry course motivated by cosmic topology. Colleagues at The College of Idaho encouraged me to teach the course repeatedly as the manuscript developed, and students there provided valuable feedback on how this story can be better told. I would like to thank Rob Beezer, David Farmer, and all my colleagues at the UTMOST 2017 Textbook Workshop and in the PreTeXt Community for helping me convert my dusty tex code to a workable PreTeXt document in order to make the text freely available online, both as a webpage and as a printable document. I also thank Jennifer Nordstrom for encouraging me to use PreTeXt in the first place. I owe my colleagues at The College of Idaho and the University of Oregon a debt of gratitude for helping to facilitate a sabbatical during which an earlier incarnation of this text developed and was subsequently published in 2009 with Jones & Bartlett. Richard Koch discussed some of its content with me, and Jim Dull patiently discussed astrophysics and cosmology with me at a moment’s notice. The pursuit of detecting the shape of the universe has rapidly evolved over the last twenty years, and Marcelo Rebouças kindly answered my questions regarding the content of Chapter 8 back in 2008, and pointed me to the latest papers. I am also grateful to the authors of the many cosmology papers accessible to the amateur enthusiast and written with the intent to inform. Finally, I would like to thank my family for their support during this endeavor, for letting this project permeate our home, and for encouraging its completion when we might have done something else, again. v vi To my son, Jasper Contents Preface iii Acknowledgements v 1 An Invitation to Geometry 1 1.1 Introduction....................1 1.2 A Brief History of Geometry..............3 1.3 Geometry on Surfaces: A First Look...........8 2 The Complex Plane 15 2.1 Basic Notions................... 15 2.2 Polar Form of a Complex Number............ 17 2.3 Division and Angle Measure.............. 19 2.4 Complex Expressions................ 22 3 Transformations 29 3.1 Basic Transformations of C .............. 29 3.2 Inversion..................... 38 3.3 The Extended Plane................. 49 3.4 Möbius Transformations............... 52 3.5 Möbius Transformations: A Closer Look......... 60 4 Geometry 71 4.1 The Basics.................... 71 4.2 Möbius Geometry.................. 78 ix x CONTENTS 5 Hyperbolic Geometry 81 5.1 The Poincaré Disk Model............... 81 5.2 Figures of Hyperbolic Geometry............ 87 5.3 Measurement in Hyperbolic Geometry.......... 91 5.4 Area and Triangle Trigonometry............ 100 5.5 The Upper Half-Plane Model............. 114 6 Elliptic Geometry 121 6.1 Antipodal Points.................. 121 6.2 Elliptic Geometry.................. 126 6.3 Measurement in Elliptic Geometry........... 132 6.4 Revisiting Euclid’s Postulates............. 142 7 Geometry on Surfaces 145 7.1 Curvature.................... 145 7.2 Elliptic Geometry with Curvature k > 0 ......... 149 7.3 Hyperbolic Geometry with Curvature k < 0 ........ 152 7.4 The Family of Geometries (Xk,Gk) ........... 158 7.5 Surfaces..................... 164 7.6 Geometry of Surfaces................ 179 7.7 Quotient Spaces.................. 184 8 Cosmic Topology 195 8.1 Three-Dimensional Geometry and 3-Manifolds....... 195 8.2 Cosmic Crystallography............... 206 8.3 Circles in the Sky.................. 214 8.4 Our Universe..................
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