Euclidean plane and its relatives

Reviewed by Susan Hagen, Senior Instructor, Virginia Tech on 2/1/18

Comprehensiveness rating: 4 see less

As the title implies, the book is a minimalist introduction to the Euclidean plane and its relatives. Much of Euclidean geometry is covered but through the lens of a Metric Space. The approach allows a faster progression through familiar Euclidean topics, but at times, that progression felt rushed. The development of the Neutral Geometry and the resulting hyperbolic plane was well written. There are other topics included such as affine, projective, and spherical geometries in the later chapters, but they are the weakest sections of the book. The last two chapters are on Geometric constructions and Area, which I would prefer to see earlier in the text. But to be fair, the book is written in a way that these chapters can be completed in different orders. There is a flow chart indicating the possible ways through the text.

Content Accuracy rating: 4

I found no mathematical errors in the text. The first chapter introduces the notation used through out the text. The explanations were clear and concise, perhaps at times too concise. The first chapter does a nice job of developing the geometry intuitively, but after formalizing the introduced observations, the rest is written succinctly. The conversational tone of the first few chapters is exchanged for the more formal approach consisting of defining objects and proving theorems about those objects. While some students will excel with the format, others will struggle with the emphasis on the symbolic notation rather than text.

Relevance/Longevity rating: 5

The book covers most of the standard geometry topics for an upper level class. The axiomatic approach to Euclidean geometry gives a more rigorous review of the geometry taught in high school.

Clarity rating: 4

The book is well written, though students may find the formal aspect of the text difficult to follow. The exercises are dispersed through out the text following the relevant theorems, with a few added “advanced exercises.”

Consistency rating: 4

The author does an excellent job using the notation established in the first chapter throughout the text. However at times a few words interspersed within the notation would help clarify the ideas for students needing more explanation.

Modularity rating: 5

The chapters are written so that once you've completed the first five chapters, you may progress through the remaining chapters in a variety of different paths, all of which are outlined in the overview using a flow chart. The sections on non-Euclidean geometry are rather short. The development of the Poincare’ disc model from inversive geometry is done very well.

Organization/Structure/Flow rating: 5

The book is well organized. Each chapter builds from the previous, with the flow chart indicating other ways to progress through the book.

Interface rating: 3

The text is a standard black and white PDF file. There are no hyperlinks such as from the table of contents to the chapters, nor from the index to the terms, but everything is written clearly.

Grammatical Errors rating: 5

I did not find grammatical errors in the text.

Cultural Relevance rating: 5

I did not notice anything that was culturally insensitive or offensive.

Comments

The geometry class I teach is required for mathematics education majors. One component I find missing is the inclusion of investigative problems using contemporary software such as GeoGebra (freeware) or Geometer’s sketchpad. However, there are some interactive java applets created using C.A.R. for students to work through on the author’s website. These problems involve completing constructions both from the Euclidean perspective, such as constructing the center of the given circle or the tangent circle to a given circle through a given point, and non-Euclidean perspective such as constructing the h-perpendicular from a given point or the tangent h-line to the given h-circle passing through a given point. Upon successful completion, students get a “Well Done” and email a screenshot of the completed work to the author.
The biggest strength of this text is the axiomatic approach used by the author to introduce the different geometries through the axioms satisfied. I would like to use the approach in my course. But given the students who populate the course, I am still trying to decide whether the book is appropriate

Reviewed by Sarah Birdsong, Lecturer, University of North Carolina at Charlotte on 2/1/18

Comprehensiveness rating: 2 see less

The textbook presents a formal axiomatic system in which classical Euclidean geometry can be interpreted. However, it doesn't provide any details or even examples of the classical (eg: Hilbert, etc) axioms. The few examples and discussions that seem, from the table of contents, to cover these standard formalizations, are treated as asides and are not given appropriate context (historical or otherwise).

The intent of the first portion of the textbook (chapters 1 through 7) appears to be to present or reinterpret classical results within the framework of a Birkhoff style axiomization. In fact, the opening of chapter 7 states,

"We discuss the most basic results in triangle geometry, mostly to show that we have developed enough machinery to prove things."

This quote refers to the theorems, lemmas, and propositions from the previous chapters. In these prior chapters, the theorems, lemmas, and propositions serve to reinterpret intuitively clear elementary facts about Euclidean geometry within the awkwardly presented Birkhoff style axiomization. It was as though the textbook wanted to be an analysis book with the concepts re-expressed in terms of classical geometry.

The level of abstraction used is unnecessary for the majority of the material covered and needlessly complicated for a standard undergraduate upper-level foundations of geometry course.

Content Accuracy rating: 2

It's difficult to fairly asses the accuracy of the textbook; the copious grammatical and spelling errors leave the reader constantly guessing what the author actually meant to convey.

Relevance/Longevity rating: 1

The textbook appears to be simply formalized lecture notes. Additionally, the manner in which they have been formalized makes the potential addition of material by people other than the author, extremely unlikely.

Clarity rating: 1

The book is riddled with grammatical and spelling errors as well as symbolic ambiguities. This makes it difficult to access the implicit content because it is unappealing to read. Moreover, the author assumes a condescending tone throughout, while at the same time omitting crucial contextual details a student would require to correctly interpret the material.

Consistency rating: 4

The non-standard notation and terminology the author presents is consistently used.

Modularity rating: 2

There are subsections and chapters as well as a given dependency graph that theoretically could be used to guide a course. However, the chapters are extraordinarily interdependent, making it difficult to assess how the chapters could stand on their own without a lot of effort on the instructor's part.

Organization/Structure/Flow rating: 1

Topics seem presented in a stream-of-conscience style and lack an overall coherence that would make the book easy for anyone to use.

Interface rating: 5

No significant formatting issues. However, the formatting used for the theorems and definitions is awkward, and some of the geometric symbols used are interpreted in a non-standard way.

Grammatical Errors rating: 1

The textbook is riddled with grammatical and spelling errors. These errors are systemic and often repeat throughout the chapters.

Cultural Relevance rating: 5

No noticeable issues.

Comments

I was considering using this book for the upper level undergraduate foundations of geometry class that I teach. However, after reading though the book, it doesn't seem as though this text would work effectively as a textbook for that course. Instead, it reads more as lecture notes or possibly as a supplemental text.