MODERN GEOMETRY I

MA 320

Catalogue Description

This course focuses on reformulation of Euclidean Geometry from an advanced viewpoint. Distance, congruences, betweeness, separation in planes and space, geometric inequalities, and the Euclidean concept of congruence without distance are covered. 3 credits. Prerequisite: Ma 192 or Equivalent.

Goals

A. To enrich a student's knowledge by learning advanced Euclidean Geometry.

B. To introduce students to non-Euclidean geometries.

C. To introduce students to the history of geometry and the influence of other cultures.

D. To provide students with an appreciation of the modeling applications of geometry.

E. To explore concepts of measurement.

F. To familiarize students with the real-world applications of geometry.

G. To provide students opportunities to develop and do proofs.

H. To provide students with the opportunity to explore one topic in depth and write a paper about it.

Procedures

A. Lecture/Discussion

B. Readings and Assigned Problems (Mainly Proofs)

C. Research Project

Course Content

A. Axiomatic Systems

1. Introduction

2. Axiomatic Systems and Their Properties

3. The Axiomatic Method

4. Models

5. Properties of Axiomatic Systems

B. Finite Geometries

1. Four Point Geometries

2. The Geometries of Fano and Young

3. Axioms for Incidence Geometry

C. Axiom Sets for Geometry

1. Euclid's Geometry and Euclid's Elements

2. An Introduction to Modern Euclidean Geometries

3. Hilbert's Model for Euclidean Geometry

4. Birkhoff's Model for Euclidean Geometry

5. The SMSG Postulates for Euclidean Geometry

6. Non Euclidean Geometries

D. Neutral Geometry

1. Introduction

2. Preliminary Notions

3. Congruence Conditions

4. The Place of Parallels

5. Saccheri-Legendre Theorem

6. The Search for a Rectangle

E. Euclidean Geometry of the Plane

1. Introduction

2. The Parallel Postulate (and Some Implications)

3. Congruence and Area

4. Similarity

5. Some Euclidean Results Concerning Circles

6. More Euclidean Results Concerning Circles

7. Some Euclidean Results Concerning Triangles

8. The Nine Point Circle

9. Measurement (standard and metric)

F. Surfaces and Solids

1. Polyhedrons, cylinders, cones, spheres

2. Volumes and surfaces

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. Project to be discussed in class - 25%

Required Text:

Bibliography

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

Alexander, Dan, and Geralyn Koeberlein, Elementary Geometry for College Students..

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.

Honors Geometry Package for Apple II 64K, IBM PC 64K. Available from Queue, Phone # 1-800-232-2224.