A Survey of Classical and Modern Geometries: With Computer Activities
Preface for the Instructor and ReaderI never intended to write a textbook and certainly not one in geometry. It was not until I taught a course to future high school teachers that I discovered that I have a view of the subject which is not very well represented by the current textbooks. The dominant trend in American college geometry courses is to use geometry as a medium to teach the logic of axiomatic systems. Though geometry lends itself very well to such an endeavor, I feel that treating it that way takes a lot of excitement out of the subject. In this text, I try to capture the joy that I have for the topic. Geometry is a fun and exciting subject that should be studied for its own sake.Though the primary target audience for this text is the future high school teacher, this text is also suitable for math majors, both because of the challenging problems throughout the text, and because of the quantity of material. In particular, I think this would make an excellent text for an undergraduate course in hyperbolic geometry.   To the StudentIn the Republic, Plato (ca. 427 - 347 B.C.) wrote that his ideal State should be ruled by philosophers educated first in mathematics. He believed that the value of mathematics is how it trains the mind, and that its practical utility is of minor importance. This philosophy is as valid now as it was then. A modern education might include vocational or technical training (such as engineering, medicine, or law), but at its core, there are the English and mathematics courses which make up a liberal education. Though mathematics has rather surprising utility, for many students, the most important lesson to be learned in their math classes is how to think analytically, creatively, and rigorously.Keep this in mind as you read this book. Recognize that the exercises are a fundamental and integral part of the text. This is where the most important lessons are learned. You will not solve them all, perhaps not even most, but I hope that the exercises you do solve will leave you with a feeling of satisfaction.   Recommended CoursesFor a college geometry course for future high school teachers, the basic course outline that I recommend and usually teach is: (Section 1.1 - 1.12: Light on Sections 1.3 and 1.4); (Section 1.13 - 1.15: Optional); (Section 3.1 - 3.7: Section 3.7 is optional); (Section 4.1 - 4.4: Integrate with Chapter 3); (Section 5.1 - 5.5: Section 5.3 is optional); (Section 6.1 - 6.2, 6.4 - 6.6: Cover quickly and sparingly); (Section 7.1-7.4, 7.6 - 7.13, 8.1 - 8.2, 8.4 - 8.5: Use an overhead).Chapter 2 on Greek astronomy provides some interesting material which can be mixed in with Chapter 1, or used on 'optional' days, such as the Wednesday before Thanksgiving. I usually begin integrating Sketchpad (Chapter 4) after I have completed the first few sections on constructions (Chapter 3). A laptop and computer projector come in handy. Polyhedra (Chapter 5) might be considered optional, but I think it can be very valuable for a future high school teacher. In particular, Exercise 5.14 should not be missed, both as a class project and again as an exercise. These are lessons which can be easily brought into the high school classroom and have the potential to be memorable. I usually skip most of Chapter 6, and only introduce the 'crutch,' the concepts of parallel and ultraparallel lines, and the concept of asymptotic triangles. The beginning of Chapter 7 poses a bit of a dilemma. Most of my students are not familiar enough with path integrals and differentials to understand the arguments of Sections 7.2 and 7.3. I could not see a way of introducing the Poincaré upper half plane model that avoids these arguments or something as difficult. I usually ask those students to accept these results and not worry too much if they do not understand the proofs. If I reach Chapter 8, it is usually covered during the last week of classes. I think of it as a cushion which allows the-students a little extra time to absorb the difficult material of Chapter 7 before their final.One of the constraints I face when I teach this course is the weak background of some of our students. Education students who have chosen mathematics as their second teaching field are required to take our geometry course. Outside of this course, the most sophisticated course they are currently required to take is the first semester of calculus. We are in the process of changing this, so that these students must also take a course in linear algebra. I think a rather nice alternative for a class of these students would be to omit Chapters 6 and 7, and instead introduce the pseudosphere (Chapter 12) as the model of hyperbolic geometry, after covering spherical geometry (Sections 10.1 - 10.5). With such a course, I would not overly emphasize the axioms of geometry. I would instead emphasize the relations between these geometries through the similar results, most notably in the different trigonometries. Such a plan would require a little more thought on the part of the instructor, since Chapter 12 was not written with this organization in mind. Nevertheless, a good instructor thoroughly familiar with the contents of Chapter 7 should be able to pull it off.   Special NotesThere are many places where the treatment of this subject could have been done differently. I would like to take a moment to explain some of my choices, as well as draw attention to and justify some of the unusual placements of material. Instructors may wish to occasionally return to this section as they teach.In Chapter 1, I never do define the measure of an angle. Though I use degrees earlier, there is no real need to talk about the measure of an angle until the Law of Cosines is introduced. Before that, for example in the Star Trek lemma, we only need a notion of congruent angles, which is defined via isometries. Since I already assume knowledge of trigonometry when I introduce the Law of Cosines, I do not see the point of formally defining the measure of angles. The student is eventually asked to formally define the measure of angles in Exercise 9.21. In Chapter 1, when we do use the measure of angles, we use degrees, which is the measure most commonly used in high schools. Later, when we introduce hyperbolic geometry, we switch to radians.There is a nice proof of Ceva's theorem (see Exercise 1.120) which does not use Menelaus' theorem. This can be used by an instructor who wishes to skip Menelaus' theorem. One advantage of the proof of Ceva's theorem using Menelaus' theorem is that it also works in both spherical and hyperbolic geometry.There is a very nice proof that cos(2π/5) is constructible (see Exercise 3.18). The advantage of the algebraic proof given in the text is that similar arguments are required in the proof that cos(2π/7) and similar quantities are not constructible.There are a number of programs similar to Geometer's Sketchpad (like Cabri and Cinderella), but I believe Sketchpad currently dominates the market, particularly in the high schools. This is why I chose to learn and write about Sketchpad.I have grown to appreciate the value of Geometer's Sketchpad and encourage instructors and readers to not just shrug off Chapter 4. It can be very useful for weak students and can be very valuable for future high school teachers. It can also be very fascinating and instructive for talented students. There are a lot of questions about constructions that I would never have considered had I not been familiar with Sketchpad. For example, which tilings of the Poincaré plane can be drawn using only a straightedge and compass? How can we construct a regular 7-gon using a straightedge, compass, and something else (see Exercise 3.39)? Some theorems, for example Feuerbach's theorem, are also a little more satisfying when played with using dynamic software (see Exercise 4.22).Results in hyperbolic trigonometry are included in Section 7.16. It is appropriate to first read about spherical trigonometry, which appears later in Sections 10.2 and 10.3. I chose to introduce hyperbolic trigonometry first only because I wanted to keep it together with the rest of Chapter 7. This could have been avoided by introducing spherical geometry first, but because we introduce new geometries via a change in Euclid's axioms, hyperbolic geometry naturally comes first.Tilings are first introduced in the exercises of Chapter 5 together with the regular and semiregular polyhedra. They are introduced again in Chapter 8, together with things of hyperbolic geometry.Chapter 9 is an unusual treatment of the foundations of geometry. It is intended for students who have already taken a course in analysis and assumes an axiomatic development of the real line.When compared to contemporary textbooks, the placement of Chapter 9 might also seem unusual, but it is not so unusual when compared with history. A sound axiomatic system for geometry was not developed until the late nineteenth century, well after the development of models for hyperbolic geometry. Though the logical order of geometry begin