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Summary: This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition. Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research. Ideal for geometry at an intermediate level.Edition/Copyright: 01
Preface for the Instructor and Reader
I 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 Student
In 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.
For 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.
There 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(2p/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(2p/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 begins with the axioms, I do not believe that it should be taught that way. A strong intuitive understanding of geometry is necessary for anyone to understand the subtleties of the axiomatic foundation.
As mentioned earlier, the placement of Chapter 10 is a matter of taste. If the instructor wishes to introduce spherical geometry earlier, there is no problem. The only prerequisites for Sections 10.1 - 10.5, other than Chapter 1, are trigonometry and some vector geometry (dot products and cross products). Parts of Chapter 5 should be done before Section 10.6, and Chapter 9 is a prerequisite for Sections 10.7 and 10.8.
If the instructor really wishes to emphasize axiomatic systems, I encourage them to look closely at Chapter 13. In this chapter, the finite affine and projective planes are first introduced as algebraic objects. 'We then define them as incidence geometries together with Desargues' theorem and eventually show that the two definitions are equivalent. This beautiful result due to Hilbert really emphasizes the relationship between algebra and geometry.
Though most of this book is meant to be read in order, there are only a few chapters which have a heavy dependence on earlier chapters. Depending on course objectives, several chapters can be safely skipped, and in particular, spherical geometry (Chapter 10) can immediately follow Chapter 1. I expect that the reader has at least a decent high school education, including trigonometry, and that they have some mathematical sophistication. I also expect that all readers cover the bulk of Chapter 1 (say, Sections 1.1 - 1.11) before moving on. More background is required for some of the text, as outlined in Table 2.
Errata and Web Support
Supporting material for this textbook will be made available at http://www.nevada.edu/~baragar/geometry.html
I anticipate that this page will include further exercises, perhaps solutions, links to related sites, and an errata sheet. Comments and reports of errors are sincerely appreciated and can be sent to email@example.com
This text evolved from a course I taught several times at the University of Nevada Las Vegas. I would like to thank all the students who took this course, and in particular, I would like to thank the class of fall '97. They showed a great deal of character by embracing this subject with nothing but classroom notes and a text we never used. Their enthusiasm was inspirational and helped motivate the creation of this text. I would like to extend special thanks to Robin Fulmer and Brenda Walker, who both lent me their notes from the fall class.
I would like to thank my advisor, Joseph Silverman, and my editor, George Lobell, whose encouragement helped transform those classroom notes into a textbook. I would like to thank Peter Shine and Dorette Pronk, who both provided feedback after they used versions of this text in courses they taught. I would like to thank Jeff Johannes, who also carefully read the text and who participated in frequent conversations about geometry and the history of mathematics. I would like to thank the reviewers too for their input. I would like to thank my production and copy editors, Barbara Mack and Martha Williams, who taught me a little about grammar.
I would like to thank John Scherk, from whom I took my first undergraduate course in geometry at the University of Alberta.
I would like to thank the members and coaches of the '98 and '99 Canadian IMO (International Mathematical Olympiad) teams. Some of the more sophisticated gems in this book are due to my association with these teams. It was also through my association with the IMO that I was exposed to Kiran Kedlaya's beautiful book Ke. I highly recommend this text to anyone with a serious interest in competition mathematics.
I would like to thank Hanns-Heinrich Langmann of Germany, Ake H. Samuelsson of Sweden, and Bogdan Enescu of Romania, for graciously allowing me to use the IMO logos from '89, '91, and '99, the years their respective countries hosted the International Mathematical Olympiad.
Finally, I would like to thank my wife Meg, and my son Timothy, whose support and tolerance made writing this text smoother and more enjoyable.
The Geometry of Our World. A Review of Terminology. Notes on Notation. Notes on the Exercises.
1. Euclidean Geometry.
The Pythagorean Theorem. The Axioms of Euclidean Geometry. SSS, SAS, and ASA. Parallel Lines. Pons Asinorum. The Star Trek Lemma. Similar Triangles. Power of the Point. The Medians and Centroid. The Incircle, Excircles, and the Law of Cosines. The Circumcircle and the Law of Sines. The Euler Line. The Nine Point Circle. Pedal Triangles and the Simson Line. Menelaus and Ceva.
2. Geometry in Greek Astronomy.
The Relative Size of the Moon and Sun. The Diameter of the Earth. The Babylonians to Kepler, a Time Line.
3. Constructions Using a Compass and Straightedge.
The Rules. Some Examples. Basic Results. The Algebra of Constructible Lengths. The Regular Pentagon. Other Constructible Figures. Trisecting an Arbitrary Angle.
4. Geometer's Sketchpad.
The Rules of Constructions. Lemmas and Theorems. Archimedes' Trisection Algorithm. Verification of Theorems. Sophisticated Results. Parabola Paper.
5. Higher Dimensional Objects.
The Platonic Solids. The Duality of Platonic Solids. The Euler Characteristic. Semiregular Polyhedra. A Partial Categorization of Semiregular Polyhedra. Four-Dimensional Objects.
6. Hyperbolic Geometry.
Models. Results from Neutral Geometry. The Congruence of Similar Triangles. Parallel and Ultraparallel Lines. Singly Asymptotic Triangles. Doubly and Triply Asymptotic Triangles. The Area of Asymptotic Triangles.
7. The Poincaré Models of Hyperbolic Geometry.
The Poincaré Upper Half Plane Model. Vertical (Euclidean) Lines. Isometries. Inversion in the Circle. Inversion in Euclidean Geometry. Fractional Linear Transformations. The Cross Ratio. Translations. Rotations. Reflections. Lengths. The Axioms of Hyperbolic Geometry. The Area of Triangles. The Poincaré Disc Model. Circles and Horocycles. Hyperbolic Trigonometry. The Angle of Parallelism. Curvature.
8. Tilings and Lattices.
Regular Tilings. Semiregular Tilings. Lattices and Fundamental Domains. Tilings in Hyperbolic Space. Tilings in Art.
Theories. The Real Line. The Plane. Line Segments and Lines. Separation Axioms. Circles. Isometries and Congruence. The Parallel Postulate. Similar Triangles.
10. Spherical Geometry.
The Area of Triangles. The Geometry of Right Triangles. The Geometry of Spherical Triangles. Menelaus' Theorem. Heron's Formula. Tilings of the Sphere. The Axioms. Elliptic Geometry.
11. Projective Geometry.
Moving a Line to Infinity. Pascal's Theorem. Projective Coordinates. Duality. Dual Conics and Brianchon's Theorem. Areal Coordinates.
12. The Pseudosphere in Lorentz Space.
The Sphere as a Foil. The Pseudosphere. Angles and the Lorentz Cross Product. A Different Perspective. The Beltrami-Klein Model. Menelaus' Theorem.
13. Finite Geometry.
Algebraic Affine Planes. Algebraic Projective Planes. Weak Incidence Geometry. Geometric Projective Planes. Addition. Multiplication. The Distributive Law. Commutativity, Coordinates, and Pappus' Theorem. Weak Projective Space and Desargues' Theorem.
The Field of Constructible Numbers. Fields as Vector Spaces. The Field of Definition for a Construction. The Regular 7-gon. The Regular 17-gon.
15. Modern Research in Geometry.
Pythagorean Triples. Bezout's Theorem. Elliptic Curves. A Mixture of Cevians. A Challenge for Fermat. The Euler Characteristic in Algebraic Geometry. Lattice Point Problems. Fractals and the Apollonian Packing Problem. Sphere Packing.
16. A Selective Time Line of Mathematics.
The Ancient Greeks. The Fifth Century A.D. to the Fifteenth Century A.D. The Renaissance to the Present.
Appendix A: Quick Reviews.
2x2 Matrices. Vector Geometry. Groups. Modular Arithmetic.
Appendix B: Hints, Answers and Solutions.
Hints to Selected Problems. Answers to Selected Problems. Solutions to Selected Problems.
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