Textbook For a Course on Classical Geometry

Question

I have been assigned to teach a first year course in geometry the next academic year. This course has been running for quite a while in the university, but of late, has been thought of as redundant because the material being covered in the course is something they have already seen in lower levels during their course of study. My task is to redesign the course. In doing so, I have essentially full freedom except for the fact that it is a first year course, and I cannot assume much in terms of prerequisites.

I want to include the proofs of anything that I cover (consistent with good pedagogy of course). I also have absolutely no problem with having to build a concept from the ground up, if it leads to an important result or is an interesting technique in its own right. My students are going to be sharp; they're some of the most gifted freshmen in the country when it comes to mathematics. I want to make this a rigorous course for them.

Can anyone suggest a good textbook that satisfies my requirements that I can follow for the course? I hope I have been (at least somewhat) clear on what I need. I will clarify anything else in the comments.

Answer 1

A fine undergraduate text is:

H. S. M. Coxeter, Introduction to Geometry

This classic work is now available in an unabridged paperback edition. The Second Edition retains all the characterisitcs that made the first edition so popular: brilliant exposition, the flexibility permitted by relatively self-contained chapters, and broad coverage ranging from topics in the Euclidean plane, to affine geometry, projective geometry, differential geometry, and topology. The Second Edition incorporates improvements in the text and in some proofs, takes note of the solution of the 4-color map problem, and provides answers to most of the exercises.

Answer 2

It depends on how much of prerequisites are to be assumed and which materials should be included. I have taught a geometry course (one semester course) for years that is aimed for sophomores. It includes Euclidean, hyperbolic and spherical geometry in a context of transformation groups (the concept of group is not introduced explicitly though). For some years, I used Geometry of Surfaces by J. Stillwell and the course covers its first four chapters. I liked the book but I felt the explanation gets sloppy in later chapters(especially chapter four). And it uses some elementary properties of complex numbers, which causes some difficulties to the students.

Later, I changed the textbook to Geometry: from Isometries to Special Relativity by N. Lee. This book is somewhat similar to the one by Stillwell but It assumes only elementary calculus as prerequisites. The first five chapters were covered in the course. The book focuses on the "three-reflection-theorem" which appears in very similar forms in all of Euclidean, hyperbolic and spherical geometry. The chapter 6 is about the geometry of special relativity, for which some background in physics can be helpful. But I have never been able to cover chapter 6 due to the time limit.

Answer 3

I recommend Geometry: A Metric Approach with Models by Richard Millman and George Parker. I have used it as a textbook in the past and liked it.

Addendum. A little bit about this book, in response to one of the comments: It offers an axiomatic and rigorous development of classical and non-Euclidean geometries. The axiomatic approach is very effective and well-explained, in my opinion. Here is a quote from the beginning of the text:

In the early development of geometry the point of view was that an axiom was a statement that described the true state of the universe. Axioms were thought of as "basic truths". Such axioms should be "self-evident". Of the basic axioms stated by Euclid in his Elements, all but one was accepted by the mathematical community as "true" and self-evident. However, his fifth axiom, which dealt with parallel lines, was not as well received. While everyone agreed it was true (whatever that meant) it was by no means obvious. For over two thousand years mathematicians tried to show that the fifth axiom was a theorem which could be proved on the basis of the remaining axioms. As we shall see, such efforts were doomed to fail. The modern view is that an axiom is a statement of a useful property. When we assume an axiom holds, usually in a definition, we are saying that we want to discuss those objects which possess this special property. We are making no statement as to whether the axiom is a statement about the real world. Rather, we are saying "accept the following as a hypothesis."

As the title suggests, the book takes the metric approach to developing and studying geometry. This approach is due to George Birkhoff. In this approach, the concept of distance (or a metric) and angle measurement is added to that of an incidence geometry to obtain basic ideas of betweenness, line segments and congruence. Another possible approach to geometry (not treated in this book) is Davis Hilbert's synthetic approach.

The book works with a few models such as the Cartesian Plane, The Poincare Upper Half Plane and The Taxicab Plane. When a new axiom is introduced, it is shown which models do (or do not) satisfy that axiom. The authors say in the Preface:

We hope that through an intimate acquaintance with examples (and a model is just an example), the reader will obtain a real feeling and intuition for non-Euclidean (and in particular, hyperbolic) geometry. From a pedagogical viewpoint this approach has the advantage of reducing the reader's tendency to reason from a picture. In addition, our students have found the strange new world of the non-Euclidean geometries both interesting and exciting.

I can confirm the last statement about students finding the strange new world of the non-Euclidean geometries both interesting and exciting. The text also has an adequate number of exercises at the end of each section.

Answer 4

Given that this is a first course, I'd use a combination of Harold Jacobs Geometry: Seeing, Doing, Understanding (first edition is my favorite) and I'd also mix in some sections of Euclid's Elements by Oliver Byrne, which is illustrated with color — see https://www.c82.net/euclid/

Lastly, I'd incorporate some philosophy that was happening at the time. Not a lot, but some interesting aspects that will help raise curiosity and wonder. It's important to put yourself in the context of what was happening in classical mathematics. It was not simply "Hey guys, let's do math!" These people were not doing math just to math it up. Classical geometry was heavily rooted to philosophy and all of what they were doing was to explore philosophical problems, not mathematical problems. I do not see a point in classical math, unless it is to include the philosophy side.

Elements is a great introduction to proofs, so I'd cherry-pick some examples that will help the student ease into proving theorems. Contemporary students cannot handle this kind of book solely. It should be used more like a snack these days. I believe it is important to remind them when they're actually looking at the Elements, to remind them of the difficulties of such a book, so that they can feel a sense of accomplishment when they do get through portions of it. Elements deserves to be mythologized and should be propped up as if it were a book written by an ancient wizard, IMO.

Answer 5

[Too long to comment]

Agreed with Krisman's comment (need to understand context better). But with that in mind, some thoughts:

1. If the target is prospective math teachers (perhaps not all stars, and some weak, but still not completely remedial as in next bullet), then I would design the course more around teaching geometry (there are texts that have this slant). Perhaps a few intuitions and insights that go beyond a classical US high school text, that might enrich their teaching...but really...not much in that direction.

2. If the target is remedial instruction, and this is then extremely remedial, then I would used something that mimics a normal US high school class. Perhaps with some packaging, like the title, so that it doesn't appear to be high school. But high school content in sheep's clothing.

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In both cases, I would be very wary of being too enriched or proof-y. It's a natural instinct of new teachers, emerging from their own research experience. We see it all the time no this forum.

specially in #2, be wary. Simple SAS two column stuff is enough.

Either case is not dealing with students that are prospective grad students. You probably already think you're aiming a notch below that, but take it down another 1-2 notches.