A Survey of Classical and Modern Geometries With Computer Activities

by Baragar, Arthur

• ISBN13: 9780130143181
• ISBN10: 0130143189
• Edition: 1st
• Format: Paperback
• Copyright: 2000-12-20
• Publisher: Pearson

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.

Table of Contents

Preface for the Instructor and Reader	ix
Introduction	1	(4)
The Geometry of Our World	1	(2)
A Review of Terminology	3	(1)
Notes on Notation	3	(1)
Notes on the Exercises	4	(1)
Euclidean Geometry	5	(54)
The Pythagorean Theorem	8	(4)
The Axioms of Euclidean Geometry	12	(3)
SSS, SAS, and ASA	15	(2)
Parallel Lines	17	(3)
Pons Asinorum	20	(1)
The Star Trek Lemma	21	(4)
Similar Triangles	25	(5)
Power of the Point	30	(4)
The Medians and Centroid	34	(2)
The Incircle, Excircles, and the Law of Cosines	36	(5)
The Circumcircle and the Law of Sines	41	(3)
The Euler Line	44	(2)
The Nine Point Circle	46	(2)
Pedal Triangles and the Simson Line	48	(5)
Menelaus and Ceva	53	(6)
Geometry in Greek Astronomy	59	(10)
The Relative Size of the Moon and Sun	60	(2)
The Diameter of the Earth	62	(2)
The Babylonians to Kepler, A Time Line	64	(5)
Constructions Using a Compass and Straightedge	69	(18)
The Rules	70	(1)
Some Examples	70	(1)
Basic Results	71	(2)
The Algebra of Constructible Lengths	73	(4)
The Regular Pentagon	77	(4)
Other Constructible Figures	81	(3)
Trisecting an Arbitrary Angle	84	(3)
Geometer's Sketchpad	87	(10)
The Rules of Constructions	87	(1)
Lemmas and Theorems	88	(3)
Archimedes' Trisection Algorithm	91	(1)
Verification of Theorems	91	(2)
Sophisticated Results	93	(3)
Parabola Paper	96	(1)
Higher Dimensional Objects	97	(18)
The Platonic Solids	97	(3)
The Duality of Platonic Solids	100	(1)
The Euler Characteristic	100	(4)
Semiregular Polyhedra	104	(4)
A Partial Categorization of Semiregular Polyhedra	108	(2)
Four-Dimensional Objects	110	(5)
Hyperbolic Geometry	115	(16)
Models	115	(3)
Results from Neutral Geometry	118	(3)
The Congruence of Similar Triangles	121	(1)
Parallel and Ultraparallel Lines	122	(3)
Singly Asymptotic Triangles	125	(1)
Double and Triply Asymptotic Triangles	126	(3)
The Area of Asymptotic Triangles	129	(2)
The Poincare Models of Hyperbolic Geometry	131	(46)
The Poincare Upper Half Plane Model	131	(1)
Vertical (Euclidean) Lines	132	(1)
Isometries	133	(1)
Inversion in the Circle	134	(8)
Inversion in Euclidean Geometry	142	(2)
Fractional Linear Transformations	144	(6)
The Cross Ratio	150	(3)
Translations	153	(3)
Rotations	156	(1)
Reflections	157	(2)
Lengths	159	(2)
The Axioms of Hyperbolic Geometry	161	(1)
The Area of Triangles	162	(3)
The Poincare Disc Model	165	(1)
Circles and Horocycles	166	(1)
Hyperbolic Trigonometry	167	(6)
The Angle of Parallelism	173	(2)
Curvature	175	(2)
Tilings and Lattices	177	(16)
Regular Tilings	177	(1)
Semiregular Tilings	178	(2)
Lattices and Fundamental Domains	180	(2)
Tilings in Hyperbolic Space	182	(6)
Tilings in Art	188	(5)
Foundations	193	(16)
Theories	193	(1)
The Real Line	194	(1)
The Plane	195	(1)
Line Segments and Lines	196	(2)
Separation Axioms	198	(2)
Circles	200	(3)
Isometries and Congruence	203	(2)
The Parallel Postulate	205	(1)
Similar Triangles	205	(4)
Spherical Geometry	209	(18)
The Area of Triangles	209	(2)
The Geometry of Right Triangles	211	(3)
The Geometry of Spherical Triangles	214	(3)
Menelaus' Theorem	217	(4)
Heron's Formula	221	(1)
Tilings of the Sphere	222	(1)
The Axioms	223	(1)
Elliptic Geometry	224	(3)
Projective Geometry	227	(20)
Moving a Line to Infinity	228	(3)
Pascal's Theorem	231	(3)
Projective Coordinates	234	(3)
Duality	237	(1)
Dual Conics and Brianchon's Theorem	238	(2)
Areal Coordinates	240	(7)
The Pseudosphere in Lorentz Space	247	(16)
The Sphere as a Foil	247	(2)
The Pseudosphere	249	(5)
Angles and the Lorentz Cross Product	254	(2)
A Different Perspective	256	(3)
The Beltrami-Klein Model	259	(1)
Menelaus' Theorem	260	(3)
Finite Geometries	263	(22)
Algebraic Affine Planes	263	(3)
Algebraic Projective Planes	266	(3)
Weak Incidence Geometry	269	(3)
Geometric Projective Planes	272	(1)
Addition	273	(3)
Multiplication	276	(1)
The Distributive Law	277	(1)
Commutativity, Coordinates, and Pappus' Theorem	278	(3)
Weak Projective Space and Desargues' Theorem	281	(4)
Nonconstructibility	285	(14)
The Field of Constructible Numbers	285	(2)
Fields as Vector Spaces	287	(4)
The Field of Definition for a Construction	291	(3)
The Regular 7-gon	294	(1)
The Regular 17-gon	295	(4)
Modern Research in Geometry	299	(30)
Pythagorean Triples	299	(3)
Bezout's Theorem	302	(3)
Elliptic Curves	305	(4)
A Mixture of Cevians	309	(2)
A Challenge for Fermat	311	(1)
The Euler Characteristic in Algebraic Geometry	312	(1)
Lattice Point Problems	313	(2)
Fractals and the Apollonian Packing Problem	315	(9)
Sphere Packing	324	(5)
A Selective Time Line of Mathematics	329	(6)
The Ancient Greeks	329	(2)
The Fifth Century A.D. to the Fifteenth Century A.D.	331	(1)
The Renaissance to the Present	331	(4)
A Quick Reviews	335	(4)
A.1 2 x 2 Matrices	335	(1)
A.2 Vector Geometry	336	(1)
A.3 Groups	336	(1)
A.4 Modular Arithmetic	337	(2)
B Hints, Answers, and Solutions	339	(24)
B.1 Hints to Selected Problems	339	(3)
B.2 Answers to Selected Problems	342	(3)
B.3 Solutions to Selected Problems	345	(18)
Bibliography	363	(3)
Index	366

Excerpts

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 theRepublic,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 Poincare 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 classe