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Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces, 2/E
David W. Henderson, Cornell University

ISBN-10: 0130309532
ISBN-13: 9780130309532

Publisher: Prentice Hall
Copyright: 2001
Format: Cloth; 386 pp
Published: 07/28/2000

This item has been replaced by Experiencing Geometry, 3/E.

For undergraduate-level courses in Geometry.

Henderson invites students to explore the basic ideas of geometry beyond the formulation of proofs. The text conveys a distinctive approach, stimulating students 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 students to gather their reasonings and understandings of each problem and discuss their findings in an open forum.

NEW - Integration of hyperbolic and spherical geometry with the Euclidean geometry—Non-Euclidean geometries are not divided into separate chapters. Every geometric notion is explored in relation to the Euclidean plane, on spheres and on hyperbolic planes.
- Allows students to learn new geometries and gain a better understanding of basic notions of Euclidean geometry. Ex.___

NEW - Use of concrete models to introduce and explore spherical geometry and hyperbolic geometry—Unique approach to hyperbolic geometry.
- Provide students with a concrete, intuitive introduction to hyperbolic geometry that gives meaning to usually abstract presentations. Ex.___

NEW - Discussion of the shape of space—Introduces the geometry needed to understand the possibilities for the shape of the physical universe.
- Allows instructors and students to appreciate the results of the new and forthcoming space observations. Ex.___

NEW - Introduction to the notion of geometric manifold—Covers both two-dimensional and three-dimensional geometric manifold.
- Exposes students to the basic idea of “manifold” which has applications in physics, engineering, biology, and cosmology. Ex.___

NEW - Organization—There is a minimum core (Chapters 1-3, 6, 8, and either 7 or 10) and the remaining 15 chapters are essentially independent of each other.
- Allows instructors to present these 15 chapters in any order after the minimum core has been covered. Ex.___

Problems based—Reflects recent NCTM and NRC standards for student-centered, activity-centered textbooks. More than 80 problems lead students to explore and experience the geometric notions in the text.
- Multi-part, open-ended problems allow students to experience modern and classical meanings and concepts of geometry, and gain experience of what it means to do mathematics through the use of multi-part, open-ended problems. Ex.___

Emphasis on the distinctions between intrinsic and extrinsic, and local and global.
- Provides students with a modern and alive approach that applies to other aspects of their lives. Ex.___

Parallel transport—Introduced early and used throughout the text.
- Introduces students to a powerful and easily understood notion that has practical applications in multiple disciplines. Ex.___

Visualization—More than 220 drawings, diagrams, and photos support the text's premise that geometry is based on visual experiences. The use of concrete models for visualization, discovery and meaning is encouraged.

Technology—An author-maintained Web page will provide links to relevant Web sites and geometric computer activities.

History—Historical discussions are interspersed throughout the text, and three chapters are largely devoted to topics of historical interest (Chapters 10, 13, 18). Also Chapter 22 is mostly devoted to current history.

Encouragement of writing—The problems can easily be used as writing assignments.
- Provides students with opportunities to improve their writing skills. Ex.___

Modern material—Includes much more modern material than other texts on the market, with more emphasis on intuition in geometry as opposed to axiomatics.

Integration of hyperbolic and spherical geometry with the Euclidean geometry—Non-Euclidean geometries are not divided into separate chapters. Every geometric notion is explored in relation to the Euclidean plane, on spheres and on hyperbolic planes.
- Allows students to learn new geometries and gain a better understanding of basic notions of Euclidean geometry. Ex.___

Use of concrete models to introduce and explore spherical geometry and hyperbolic geometry—Unique approach to hyperbolic geometry.
- Provide students with a concrete, intuitive introduction to hyperbolic geometry that gives meaning to usually abstract presentations. Ex.___

Discussion of the shape of space—Introduces the geometry needed to understand the possibilities for the shape of the physical universe.
- Allows instructors and students to appreciate the results of the new and forthcoming space observations. Ex.___

Introduction to the notion of geometric manifold—Covers both two-dimensional and three-dimensional geometric manifold.
- Exposes students to the basic idea of “manifold” which has applications in physics, engineering, biology, and cosmology. Ex.___

Organization—There is a minimum core (Chapters 1-3, 6, 8, and either 7 or 10) and the remaining 15 chapters are essentially independent of each other.
- Allows instructors to present these 15 chapters in any order after the minimum core has been covered. Ex.___

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.

Experiencing Geometry, 3/E
Henderson & Taimina
©2005 | Prentice Hall | Paper; 432 pp | Instock
ISBN-10: 0131437488 | ISBN-13: 9780131437487
Brief Description | Buy from myPearsonStore

For junior/senior level course in Geometry.

The distinctive approach of Henderson and Taimina's text stimulates students to develop a broader, deeper understanding of mathematics through active experience—including discovery, discussion, writing fundamental ideas and learning about the history of those ideas. A series of interesting problems (ranging from easy to challenging) encourage students to gather and discuss their reasonings and understanding. This is the only undergraduate text that pays attention to geometric intuition, student cognitive development, and rigorous mathematics; and that includes a broad vision of geometry that leads to discussion of four strands in the history of geometry and to an understanding of the possible shapes of the physical universe.

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