Spherical Geometry and Its Applications

The delights and uses of spherical geometry should be more well known.  Where else can you find an easily accessible geometry found in nearly every sports arena and garage, that physically surrounds us! This geometry has been a professional love of mine for many years, and I was excited to review this text. I hoped that it would be something to share with my students, but found instead that it is a worthy way for curious and initiated mathematicians to learn some interesting spherical geometry and explore its lovely applications (pointed out in the last paragraph of this review).

 

Whittlesey’s intent is clearly stated in the preface: “This work is an attempt to bring a comprehensive coverage of spherical geometry and its applications to a modern audience that is unfamiliar with it.” (p. vii). The author then makes the good points that spherical geometry has not been regularly taught in high schools (or at the collegiate level) in the US since the 1950s and that it has many useful applications in astronomy, geology, and crystallography among others. The author also makes clear that “This book is intended as a course in spherical geometry for mathematics majors.” (p. viii). Therefore, I decided to evaluate this text as one that I might adopt for a Non-Euclidean geometry course or use as a supplement for a Euclidean course.

 

With my students in mind, I eagerly began to dive into this book. What I found was often disappointing: some of the pedological choices didn’t make sense, there were many visual distractors and there was a general lack of attention to detail.

 

The first two chapters seemed to be an unfruitful tug-of-war between two different approaches to geometry: analytic and synthetic. I understand why these choices were made. There are two different directions the author is attempting to head: toward an axiomatic development of spherical geometry and using it in applications. This means there is a lot of Euclidean background that students will need in order to be successful understanding later material. To combat this overload of possibly necessary prerequisites, I wish the author had provided suggestions for different tracks on how to use this text: one focusing on axiomatic development and another focusing on applications.

 

Chapter 3 is long, ambitious, and meaty, attempting to develop spherical geometry from a set of axioms. Spherical geometry is a challenge to rigorously tackle in this way! The facts of antipodal points and that there are at least two great circle line segments between every pair of points defy simple axioms. Whittlesey attempts to solve some of these challenges with an axiom that sets up a one-to-one correspondence between points on a great circle and \( \mathbb{R}/2\pi \) (\( \mathbb{R} \) modulo  \( 2 \pi \)) involving equivalence classes and a distance function. But consider this definition and think about the notation, mathematical maturity, and background necessary for a student to fully comprehend it.

 

 

I know that my students would really struggle with all the layers in this definition. Given the author’s approach, however, this is a necessary and fundamental definition. While there are valuable insights to be gained here, there are simpler approaches (see An Introduction to Non-Euclidean Geometry by David Gans).

 

The notation for rays and other geometric objects also didn’t make pedagogical sense to me and seemed visually distracting. I understand that the author did this to be precise, but it adds an unnecessary layer to decode when, in most cases, the meaning is clear from context.  Unfortunately, this is only one aspect of the visual distractions that a new learner would encounter. In places, the typesetting and figures are unnecessarily cluttered, opaque, and nonsensical. For example, each axiom comes with both a number from the surrounding text (like “A-12”) and a number from its place in the list of axioms (like “Proposition 15.1”). The author never uses the “A-#” notation and worse, refers to at least two axioms as propositions.  For example, Axiom 12 and Proposition 15.1 refer to the same statement which would not help a learner keep the roles of propositions and axioms separate. Similar pedagogically questionable choices also occur in the remaining chapters on applications. All of this points to a lack of attention to detail in preparing the book and makes this feel like a collection of notes intended to support a series of lectures rather than a coherent whole meant to lead students to a solid understanding.

 

However, all is not lost! While the pedological uses of this text are limited, there is another set of strengths that could be appreciated by a different audience. Whittlesey does a fantastic job of translating a selection of ideas from astronomy, crystallography, and geology (the applications) into the language and notation that is commonly accepted and used in the broader mathematical community. The author is able to bridge this divide and provide an opportunity for those steeped in the broader mathematical community’s notations, conventions, and language to enter those applications and to see the use of spherical geometry. It is perhaps a Rosetta Stone that will entice mathematicians to explore those applications. The reverse might also be true; this text might enable a specialist in one of those application areas to learn the mathematical community’s notations and conventions and further explore spherical geometry.

 

I think Whittlesey has at least partially achieved the goal of bringing “comprehensive coverage of spherical geometry and its applications to a modern audience”, just not as a textbook for undergraduates. An experienced mathematician willing to overlook typesetting and other difficulties will find this a pleasurable book to read. There are so many fascinating aspects of and uses of spherical geometry to ponder: What is the history of this geometry? (Endnotes after the questions in most sections), How can you axiomatize spherical geometry? (Chapter 3), How do you figure out when an object rises or sets in the sky? (Section 25) , What are the angles between the faces of a triacontahedron and what is it? (Problem 16 on Page 212), How can you use spherical geometry to explain Bragg’s law in crystallography? (Section 27), How can you use isometries to compute the average slip rate of a geological fault? (Page 231), How can I use quaternions to derive the basic trigonometric relations between the sides and angles of spherical triangle thus converting what has historically been a non-obvious route through a trigonometric briar patch into a more elegant and methodical way? (Chapter 8). Read this text to find out!

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William Dickinson has loved geometry since eighth grade when he became fascinated with trisecting the angle using only a straight edge and compass and he has been sharing that love with his students since arriving a Grand Valley State University in 2000. It is his fundamental belief that every Euclidean result has an analogous one in spherical and hyperbolic geometry and he would love to know the analog of Morley’s Trisector Theorem!  His latest (pandemic) project has centered on rewriting Spherical Easel, an application for exploring spherical geometry, which he first co-authored in 2002, into a modern web-based app: https://easelgeo.app.