MODERN GEOMETRY II

MA 321

Catalogue Description

The three geometries, parallel postulates and projections, area functions, perpendicular lines and planes, Saccheri quadrilaterals, inversion and reflection, hyperbolic geometry, absolute geometry, Bolyai’s theorem, defects and area, and consistency of the hyperbolic postulates are studied in this course. 3 credits. Prerequisite: Ma 320 Modern Geometry I or Equivalent.

Goals

A. To enrich a student's knowledge by learning Non-Euclidean Geometries and their implications and consequences.

Procedures

A. Lecture/Discussion

B. Readings and Assigned Problems (Mainly Proofs)

C. Research Project

Course Content

A. Side Trips - Analytic Geometry

1. Introduction

2. Analytic Geometry

3. Historical Perspectives

4. Coordination of the Plane

5. Distance in the Plane

6. Analytic Equations of Straight Lines and Circles

7. Applications of Analytical Geometry

B. Side Trips - Transformational Geometry

1. Transformational Geometry

1. Introduction

2. Mappings and Transformations

3. Isometries

4. Applications of Isometries to Theorem Proving

5. Similarities and Their Applications to Theorem Proving

2. Analytic Transformations

1. Introduction

2. Analytical Equations for Isometries

3. Analytical Equations for Similarities

4. Applications of Isometries and Similarities Using Analytical Transformations.

3. Inversion

1. Introduction

2. Inversion in a Circle

C. Non-Euclidean Geometries

1. Non-Euclidean Geometries

1. Introduction

2. A Return to Neutral Geometry - The Angle of Parallelism

3. The Hyperbolic Parallel Postulate

4. Some Hyperbolic Results Concerning Polygons

5. Area in Hyperbolic Geometry

6. Showing Consistency - A Model for Hyperbolic Geometry

7. Classifying Theorems

2. Elliptical Geometry

1. A Geometry with no Parallels

2. Two Models

3. Some Results in Elliptic Geometry

3. Geometry in the Real World

D. Projective Geometry

1. Introduction

2. The Real Projective Plane

3. Duality

4. Perspectivity

5. Theorem of Desargues

6. Projective Transformations

Evaluation Methods

1. 3 Exams. There may be a take home exam due to nature of material - 50%

2. Final 2 hour Comprehensive Exam covering the term - 25%

3. Research Project - 25%

Bibliography

Required Text: Wallace, Edward C. & West, Stephen F., Roads to Geometry, Prentice Hall, Englewood Cliffs, N.J., 1992.

Adler, Claire Fisher, Modern Geometry, An Integrated First Course, 2nd Ed., McGraw Hill Publishing, New York, 1967.

Coxeter, H.S.M., Introduction to Geometry, 2nd Ed., John Wiley & Sons, New York, 1969.

Eves, Howard A., Survey of Geometry, Rev. Ed., Allyn & Bacon, Boston, Mass., 1972.

Fishback, W.T., Projective and Euclidean Geometry, 2nd Ed., John Wiley & Sons, New York, 1969.

Greenberg, Marvin Jay, Euclidean and Non Euclidean Geometries, Development and History, 2nd Ed., W.H. Freeman & Co., New York, 1980.

Jacobs, Harold R., Geometry, 2nd Ed., W.H. Freeman & Co., New York, 1987.

Moise, Edwin E., Elementary Geometry From an Advanced Standpoint, 3rd Ed., Addison-Wesley, New York, 1990.

Posamentier, Alfred S., Excursions in Advanced Euclidean Geometry, Rev. Ed., Janson Publishing (Addison-Wesley), Providence, Rhode Island, 1984. (Chapters 1-3)

Rich, Barnett, Theory and Problems of Geometry, Schaum's Outline Series, 2nd Ed., McGraw Hill, New York, 1989.

Smart, James R., Modern Geometries, 3rd Ed., Brooks/Cole Publishing, California, 1988.

Software

The Geometric Supposers. Designed by: Education Development Center, Dr. Judah L. Schwartz, MIT/Harvard, and Dr. Michal Yerushalmy, EDC. Available from Sunburst Communications.