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Experiencing Geometry : In Euclidean, Spherical and Hyperbolic Spaces / Edition 2

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Overview

The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers to gather their reasonings and understandings of each problem.
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Product Details

  • ISBN-13: 9780130309532
  • Publisher: Pearson Education
  • Publication date: 7/28/2000
  • Edition description: Older Edition
  • Edition number: 2
  • Pages: 352
  • Product dimensions: 7.01 (w) x 9.25 (h) x 0.30 (d)

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PREFACE:

PREFACE

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the productions of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth — the very experience out of which Love rises.
- Plotinus, The Enneads, 11.9.16 A: Plotinus

This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere. The most important change is that I have included material on hyperbolic geometry that was missing in the first book. This has also necessitated more discussions of circles and their properties. In addition, there is added material on geometric manifolds and the shape of space. I decided to include hyperbolic geometry for two reasons: 1) the cosmologists say that our physical universe very likely has (at least in part) hyperbolic geometry, and 2) Daina Taimina, a mathematician at the University of Latvia and now my wife, figured out how to crochet a hyperbolic plane, which allowed us to explore intuitively for the first time the geometry of the hyperbolic plane. In addition, Daina Taimina has been responsible for including in this edition significantly more historical material. In this historical material we discuss and try to clear up many current misconceptions that are commonly heldabout some mathematical ideas.

This book is based on a junior/senior-level course I have been teaching since 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication. Certain problems and sections in this book require from the reader a background more advanced than first-semester calculus. These sections are indicated with an asterisk (*) and the background required is indicated (usually at the beginning of the chapter).

The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems, and are encouraged to write and speak their seasonings and understandings. I listen to and critique their thinking and use it to stimulate whole class discussions.

The formal expression of "straightness" is a very difficult formal area of mathematics. However, the concept of "straight" an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the (Euclidean) plane, on a sphere, or on a hyperbolic plane. We will study these ideas and questions, as much as is possible, from an intrinsic point-of-view — that is, the point-of-view of a 2-dimensional bug crawling around on the surface. This will lead to the question: "What is the shape of our physical three-dimensional universe?" Here we are like 3-dimensional bugs who can only view the universe intrinsically.

Most of the problems are approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). I find that by exploring the geometry of a sphere and a hyperbolic plane my students gain a deeper understanding of the geometry of the (Euclidean) plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. I find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. (All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.)

USEFUL SUPPLEMENTS

A faculty member may obtain from the publisher the Instructor's Manual (containing possible solutions to each problem and discussions on how to use this book in a course) by sending a request via e-mail to George_Lobell@prenhall.com or calling 1-201-236-7407.

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase Lénárt Sphere® sets — a transparent sphere, a spherical compass, and a spherical "straight edge" that doubles as a protractor. They work well for small group explorations in the classroom and are available from Key Curriculum Press. However, considerably less expensive alternatives are available: A beach ball or basketball will work for classroom demonstrations, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles. Also, many craft stores carry inexpensive plastic spheres that can be used successfully.

I strongly urge that you have a hyperbolic surface such as those described in Chapter 5. Unfortunately, such hyperbolic surfaces are not readily available commercially. However, directions for making such surfaces (out of paper or by crocheting) are contained in Chapter 5, and I will list patterns for making paper models and sources for crocheted hyperbolic surfaces at:

www.math.cornell.edu/~dwh/books/eg00/supplements.html

as they become available. Most books that explore hyperbolic geometry do so by considering only one of the various "models" of hyperbolic geometry, which give representations of hyperbolic geometry in the same way that a map of a portion of the earth gives a representation of a portion of the earth. Each of these representations necessarily (see Chapter 16) distorts either straight lines or angles or both.

In addition, the use of dynamic geometry software such as Geometers Sketchpad®, Cabri®, or Cinderella® will enhance any geometry course. These software packages were originally written for exploring Euclidean plane geometry, but recent versions allow one to also dynamically explore spherical and hyperbolic geometries. I will maintain at the web address listed above links to information about these software packages and to web pages that give examples on how to use them for self-learning or in a classroom.

MY TEACHING BACKGROUND

My teaching is a product of Western Civilization. My known ancestors lived in England, Scotland, Ireland, Germany, and Luxembourg and I am a descendent from a long line of academics stretching back (according to family traditions) to at least the seventeenth century. My mode of teaching also has deep Western roots that reach back to the Socratic dialogues recorded by Plato in ancient Greece. More directly, my teaching has been influenced by my experiences in high school, college, and graduate school. In Ames, Iowa, my high school world literature teacher, Mary McNally, coaxed deep creative thinking out of us through her many writing assignments which she read with great interest in our ideas. At Swarthmore College in Pennsylvania instead of classes I spent my last two years in student participation seminars and tutorials, where I learned to take charge of my own learning and become an academic scholar. In graduate school at the University of Wisconsin my mentor, R.H. Bing, taught without lectures or textbooks in a style which is often known as the Moore Method, named after Bing's graduate mentor R.L. Moore at the University of Texas. (See TG: Traylor for more information on the Moore Method.) R.L. Moore received his PhD at the University of Chicago before the turn of the century and was one of the very first Americans to receive a PhD in mathematics in this country. My teaching of the geometry course and the writing of this book evolved from this background.

David W. Henderson
Ithaca, NY

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Table of Contents

1. What Is Straight?
Problem 1.1: When Do You Call a Line Straight? How Do You Construct a Straight Line? The Symmetries of a Line. Local (and Infinitesimal) Straightness.

2. Straightness on Spheres.
Problem 2.1: What Is Straight on a Sphere? Symmetries of Great Circles. Every Geodesic Is a Great Circle. Intrinsic Curvature.

3. What Is an Angle?
Problem 3.1: Vertical Angle Theorem (VAT). Problem 3.2: What Is an Angle? Hints for Three Different Proofs. Problem 3.3: Duality Between Points and Lines.

4. Straightness on Cylinders and Cones.
Problem 4.1: Straightness on Cylinders and Cones. Cones with Varying Cone Angles. Geodesics on Cylinders. Geodesics on Cones. Locally Isometric. Is "Shortest" Always "Straight"? Relations to Differential Geometry.

5. Straightness on Hyperbolic Planes.
A Short History of Hyperbolic Geometry. Constructions of Hyperbolic Planes. Hyperbolic Planes of Different Raddi (Curvature). Problem 5.1: What Is Straight in a Hyperbolic Plane? Problem 5.2: The Pseudosphere Is Hyperbolic. Problem 5.3: Rotations and Reflections on Surfaces.

6. Triangles and Congruencies.
Geodesics are Locally Unique. Problem 6.1: Properties of Geodesics. Problem 6.2: Isosceles Triangle Theorem (ITT). Circles. Problem 6.3: Bisector Constructions. Problem 6.4: Side-Angle-Side (SAS).Problem 6.5: Angle-Side-Angle (ASA).

7. Area and Holonomy.
Problem 7.1: The Area of a Triangle on a Sphere. Problem 7.2: Area of Hyperbolic Triangles. Problem 7.3: Sum of the Angles of a Triangle. Introduction Parallel Transport and Holonomy. Problem 7.4: The Holonomy of a Small Triangle. The Gauss-Bonnet Formula for Triangles. Problem 7.5: Gauss-Bonnet Formula for Polygons. Gauss-Bonnet Formula for Polygons on Surfaces.

8. Parallel Transport.
Problem 8.1: Euclid's Exterior Angle Theorem (EEAT). Problem 8.2: Symmetries of Parallel Transported Lines. Problem 8.3: Transversals through a Midpoint. Problem 8.4: What is "Parallel"?

9. SSS, ASS, SAA and AAA.
Problem 9.1: Side-Side-Side (SSS). Problem 9.2: Angle-Side-Side (ASS). Problem 9.3: Side-Angle-Angle (SAA). Problem 9.4: Angle-Angle-Angle (AAA).

10. Parallel Postulates.
Parallel Lines on the Plane are Special. Problem 10.1: Parallel Transport on the Plane. Problem 10.2: Parallel Postulates Not Involving (Non-) Intersecting Lines). Equidistant Curves on Spheres and Hyperbolic Planes. Problem 10.3: Parallel Postulates Involving (Non-) Intersecting Lines. Problem 10.4: EFP and PPP on Sphere and Hyperbolic Plane. Comparisons of Plane, Spheres, and Hyperbolic Planes. Some Historical Notes on the Parallel Postulates.

11. Isometries and Patterns.
Problem 11.1: Isometries. Symmetries and Patterns. Problem 11.2: Examples of Patterns. Problem 11.3: Isometry Determined by Three Points. Problem 11.4: Classification of Isometries. Problem 11.5: Classification of Discrete Strip Patterns. Problem 11.6: Classification of Finite Plane Patterns. Problem 11.7: Regular Tilings with Polygons. Geometric Meaning of Abstract Group Terminology.

12. Dissection Theory.
What is Dissection Theory? Problem 12.1: Dissect Plane Triangle and Parallelogram. Dissection Theory on Spheres and Hyperbolic Planes. Problem 12.2: Khayyam Quadrilaterals. Problem 12.3: Dissect Spherical and Hyperbolic Triangles and Khayyam Parallelograms. Problem 12.4: Spherical Polygons Dissect to Lunes.

13. Square Roots, Pythagoras and Similar Triangles.
Square Roots. Problem 13.1: A Rectangle Dissects into a Square. Baudhayana's Sulbasutram. Problem 13.2: Equivalence of Squares. Any Polygon Can Be Dissected into a Square. Problem 13.3: Similar Triangles. Three-Dimensional Dissections and Hilbert's Third Problem.

14. Circles in the Plane.
Problem 14.1: Angles and Power Points of Plane Circles. Problem 14.2: Inversions in Circles. Problem 14.3: Applications of Inversions.

15. Projection of a Sphere onto a Plane.
Problem 15.1: Charts Must Distort. Problem 15.2: Gnomic Projection. Problem 15.3: Cylindrical Projection. Problem 15.4: Stereographic Projection.

16. Projections (Models) of Hyperbolic Planes.
Problem 16.1: The Upper Half Plane Model. Problem 16.2: Upper Half Plane Is Model of Annular Hyperbolic Plane. Problem 16.3: Properties of Hyperbolic Geodesics. Problem 16.4: Hyperbolic Ideal Triangles. Problem 16.5: Poincaré Disk Model. Problem 16.6: Projective Disk Model.

17. Geometric 2-Manifolds and Coverings.
Problem 17.1: Geodesics on Cylinders and Cones. n-Sheeted Coverings of a Cylinder. n-Sheeted (Branched) Coverings of a Cone. Problem 17.2: Flat Torus and Flat Klein Bottle. Problem 17.3: Universal Covering of Flat 2-Manifolds. Problem 17.4: Spherical 2-Manifolds. Coverings of a Sphere. Problem 17.5: Hyperbolic Manifolds. Problem 17.6: Area, Euler Number, and Gauss-Bonnet. Triangles on Geometric Manifolds. Problem 17.7: Can the Bug Tell Which Manifold?

18. Geometric Solutions of Quadratic and Cubic Equations.
Problem 18.1: Quadratic Equations. Problem 18.2: Conic Sections and Cube Roots. Problem 18.3: Roots of Cubic Equations. Problem 18.4: Algebraic Solution of Cubics. So What Does This All Point To?

19. Trigonometry and Duality.
Problem 19.1: Circumference of a Circle. Problem 19.2: Law of Cosines. Problem 19.3: Law of Sines. Duality on a Sphere. Problem 19.4: The Dual of a Small Triangle. Problem 19.5: Trigonometry with Congruences. Duality on the Projective Plane. Problem 19.6: Properties on the Projective Plane. Perspective Drawings and Vision.

20. 3-Spheres and Hyperbolic 3-Spaces.
Problem 20.1: Explain 3-Space to 2-D Person. Problem 20.2: A 3-Sphere in 4-Space. Problem 20.3: Hyperbolic 3-Space, Upper Half Space. Problem 20.4: Disjoint Equidistant Great Circles. Problem 20.5: Hyperbolic and Spherical Symmetries. Problem 20.6: Triangles in 3-Dimensional Spaces.

21. Polyhedra.
Definitions and Terminology. Problem 21.1: Measure of a Solid Angle. Problem 21.2: Edges and Face Angles. Problem 21.3: Edges and Dihedral Angles. Problem 21.4: Other Tetrahedra Congruence Theorems. Problem 21.5: The Five Regular Polyhedra.

22. 3-Manifolds—The Shape of Space.
Space as an Oriented Geometric 3-Manifold. Problem 22.1: Is Our Universe Non-Euclidean? Problem 22.2: Euclidean 3-Manifolds. Problem 22.3: Dodecahedral 3-Manifolds. Problem 22.4: Some Other Geometric 3-Manifolds. Cosmic Background Radiation. Problem 22.5: Circle Patterns Show the Shape of Space.

Appendix A—Euclid's Definitions, Postulates, and Common Notions.
Definitions. Postulates. Common Notions.

Appendix B—Square Roots in the Sulbasutram.
Introduction. Construction of the Savisesa for the Square Root of Two. Fractions in the Sulbasutram. Comparing with the Divide-and-Average (D&A) Method. Conclusions.

Annotated Bibliography.
AT: Ancient Texts. CG: Computers and Geometry. DG: Differential Geometry. Di: Dissections. DS: Dimensions and Scale. GC: Geometry in Different Cultures. Hi: History. MP: Models, Polyhedra. Na: Nature. NE: None-Euclideam Geometries (Mostly Hyperbolic). Ph: Philosophy. RN: Real Numbers. SE: Surveys and General Expositions. SG: Symmetry and Groups. SP: Spherical and Projective Geometry. TG: Teaching Geometry. Tp: Topology. Tx: Geometry Texts. Un: The Physical Universe. Z: Miscellaneous.

Index.
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Preface

PREFACE:

PREFACE

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the productions of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth — the very experience out of which Love rises.
- Plotinus, The Enneads, 11.9.16 A: Plotinus

This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere. The most important change is that I have included material on hyperbolic geometry that was missing in the first book. This has also necessitated more discussions of circles and their properties. In addition, there is added material on geometric manifolds and the shape of space. I decided to include hyperbolic geometry for two reasons: 1) the cosmologists say that our physical universe very likely has (at least in part) hyperbolic geometry, and 2) Daina Taimina, a mathematician at the University of Latvia and now my wife, figured out how to crochet a hyperbolic plane, which allowed us to explore intuitively for the first time the geometry of the hyperbolic plane. In addition, Daina Taimina has been responsible for including in this edition significantly more historical material. In this historical material we discuss and try to clear up many current misconceptions that are commonlyheldabout some mathematical ideas.

This book is based on a junior/senior-level course I have been teaching since 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication. Certain problems and sections in this book require from the reader a background more advanced than first-semester calculus. These sections are indicated with an asterisk (*) and the background required is indicated (usually at the beginning of the chapter).

The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems, and are encouraged to write and speak their seasonings and understandings. I listen to and critique their thinking and use it to stimulate whole class discussions.

The formal expression of "straightness" is a very difficult formal area of mathematics. However, the concept of "straight" an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the (Euclidean) plane, on a sphere, or on a hyperbolic plane. We will study these ideas and questions, as much as is possible, from an intrinsic point-of-view — that is, the point-of-view of a 2-dimensional bug crawling around on the surface. This will lead to the question: "What is the shape of our physical three-dimensional universe?" Here we are like 3-dimensional bugs who can only view the universe intrinsically.

Most of the problems are approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). I find that by exploring the geometry of a sphere and a hyperbolic plane my students gain a deeper understanding of the geometry of the (Euclidean) plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. I find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. (All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.)

USEFUL SUPPLEMENTS

A faculty member may obtain from the publisher the Instructor's Manual (containing possible solutions to each problem and discussions on how to use this book in a course) by sending a request via e-mail to George_Lobell@prenhall.com or calling 1-201-236-7407.

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase Lénárt Sphere® sets — a transparent sphere, a spherical compass, and a spherical "straight edge" that doubles as a protractor. They work well for small group explorations in the classroom and are available from Key Curriculum Press. However, considerably less expensive alternatives are available: A beach ball or basketball will work for classroom demonstrations, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles. Also, many craft stores carry inexpensive plastic spheres that can be used successfully.

I strongly urge that you have a hyperbolic surface such as those described in Chapter 5. Unfortunately, such hyperbolic surfaces are not readily available commercially. However, directions for making such surfaces (out of paper or by crocheting) are contained in Chapter 5, and I will list patterns for making paper models and sources for crocheted hyperbolic surfaces at:

www.math.cornell.edu/~dwh/books/eg00/supplements.html

as they become available. Most books that explore hyperbolic geometry do so by considering only one of the various "models" of hyperbolic geometry, which give representations of hyperbolic geometry in the same way that a map of a portion of the earth gives a representation of a portion of the earth. Each of these representations necessarily (see Chapter 16) distorts either straight lines or angles or both.

In addition, the use of dynamic geometry software such as Geometers Sketchpad®, Cabri®, or Cinderella® will enhance any geometry course. These software packages were originally written for exploring Euclidean plane geometry, but recent versions allow one to also dynamically explore spherical and hyperbolic geometries. I will maintain at the web address listed above links to information about these software packages and to web pages that give examples on how to use them for self-learning or in a classroom.

MY TEACHING BACKGROUND

My teaching is a product of Western Civilization. My known ancestors lived in England, Scotland, Ireland, Germany, and Luxembourg and I am a descendent from a long line of academics stretching back (according to family traditions) to at least the seventeenth century. My mode of teaching also has deep Western roots that reach back to the Socratic dialogues recorded by Plato in ancient Greece. More directly, my teaching has been influenced by my experiences in high school, college, and graduate school. In Ames, Iowa, my high school world literature teacher, Mary McNally, coaxed deep creative thinking out of us through her many writing assignments which she read with great interest in our ideas. At Swarthmore College in Pennsylvania instead of classes I spent my last two years in student participation seminars and tutorials, where I learned to take charge of my own learning and become an academic scholar. In graduate school at the University of Wisconsin my mentor, R.H. Bing, taught without lectures or textbooks in a style which is often known as the Moore Method, named after Bing's graduate mentor R.L. Moore at the University of Texas. (See TG: Traylor for more information on the Moore Method.) R.L. Moore received his PhD at the University of Chicago before the turn of the century and was one of the very first Americans to receive a PhD in mathematics in this country. My teaching of the geometry course and the writing of this book evolved from this background.

David W. Henderson
Ithaca, NY

Read More Show Less

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