Geometry

with an Introduction to Cosmic Topology

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Mike Hitchman's open-content text 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 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, over 200 exercises, and nearly as many figures. Dynamic geometry activities complement the material as well.

Read more about the text below, or get right to it.

Previously published by Jones & Bartlett in 2009 under the cover at right, the author now happily makes an updated and revised edition freely available here, thanks in large part to the Mathbook XML Project.


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You can read the online version right now, or download a corresponding static document. An inexpensive bound copy will be available soon if you would like to have a book to hold in your hands or to have on your shelf.


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Figures in the text

The vast majority of the figures were originally generated using The Geometer's Sketchpad. In the present edition, most figures have been created in LaTeX as TikZ images. If you are interested in obtaining the code for these images, feel free to contact the author.

More about the text

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, elliptic, and Euclidean. These are the geometries we study in this text.

Chapters 2 through 7 contain the core mathematical content of the text. We follow 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 hypberbolic and elliptc geometry in Chapters 5 and 6, respectfully. 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 develops 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 geometric perspective taken in this text. In my experience it is very helpful to spend time discussing this content in class. The Coneland and Saddleland exercises in Chapter 1 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 are 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 brief 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 differential 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.

Contact

Mike Hitchman
Department of Mathematics
Linfield College
mhitchm (at) linfield (dot) edu


August, 2017

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