A Course in Modern Geometries pp 1-32 | Cite as

# Axiomatic Systems and Finite Geometries

## Abstract

Finite geometries were developed in the late nineteenth century, in part to demonstrate and test the axiomatic properties of *completeness, consistency*, and *independence*. They are introduced in this chapter to fulfill this historical role and to develop both an appreciation for and an understanding of the revolution in mathematical and philosophical thought brought about by the development of non-Euclidean geometry. In addition, finite geometries provide relatively simple axiomatic systems in which we can begin to develop the skills and techniques of geometric reasoning. The finite geometries introduced in Sections 1.3 and 1.5 also illustrate some of the fundamental properties of non-Euclidean and projective geometry.

## Keywords

Projective Plane Distinct Point Projective Geometry Axiomatic System Code Word## Preview

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## Suggestions for Further Reading

- Albert, A. A., and Sandler, R. (1968).
*An Introduction to Finite Projective Planes*. New York: Holt, Rinehart and Winston. (Contains a thorough group theoretic treatment of finite projective planes.)zbMATHGoogle Scholar - Anderson, I. (1974).
*A First Course in Combinatorial Mathematics*. Oxford, England: Clarendon Press. (Chapter 6 discusses block designs and error-correcting codes.)zbMATHGoogle Scholar - Beck, A., Bleicher, M. N., and Crowe, D. W. (1972).
*Excursions into Mathematics*. New York: Worth. (Sections 4.9–4.15 give a very readable discussion of finite planes, including the development of analytic models.)Google Scholar - Benedicty, M., and Sledge, F. R. (1987).
*Discrete Mathematical Structures*. Orlando, FL: Harcourt Brace Jovanovich. (Chapter 13 gives an elementary presentation of coding theory.)Google Scholar - Cipra, B. A. (1988). Computer search solves old math problem.
*Science*242:1507–1508. (Reports on verification that there is no projective plane of order 10.)MathSciNetCrossRefGoogle Scholar - Gallian, J. (1996). Error detection methods.
*ACM Computing Surveys*. Vol. 28, No. 3, pp. 504–517.CrossRefGoogle Scholar - Gensler, H. J. (1984).
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*Mathematics: People, Problems, Results*. D. M. Campbell and J. C. Higgins (Eds.), Vol. 2, pp. 262–275. Belmont, CA: Wadsworth.Google Scholar - Kolata, G. (1982). Does Gödel’s theorem matter to mathematics?
*Science*218: 779–780.MathSciNetzbMATHCrossRefGoogle Scholar - Lam, C. W. H. (1991). The Search for a Projective Plane of Order 10.
*The American Mathematical Monthly*. Vol. 98, No. 4, pp. 305–318.MathSciNetzbMATHCrossRefGoogle Scholar - Lockwood, J. R., and Runion, G. E. (1978).
*Deductive Systems: Finite and Non-Euclidean Geometries*. Reston,VA: NCTM (Chapter 1 contains an elementary discussion of axiomatic systems.)Google Scholar - Nagel, E. and Newman, J. R. (1956). Gödel’s proof. In
*The World of Mathematics*. J. R. Newman (Ed.), Vol. 3, pp. 1668–1695. New York: Simon and Schuster.Google Scholar - Pless, V. (1982).
*Introduction to the Theory of Error-Correcting Codes*. New York: Wiley. (A well-written explanation of this new discipline and the mathematics involved.)zbMATHGoogle Scholar - Smart, J. R. (1998).
*Modern Geometries*, 5th ed. Pacific Grove, CA: Brooks/Cole. (Chapter 1 contains an easily readable discussion of axiomatic systems and several finite geometries.)Google Scholar - Thompson, T. M. (1983).
*From Error-Correcting Codes Through Sphere Packings to Simple Groups*. The Carus Mathematical Monographs, No. 21. Ithaca, NY: MAA (Incorporates numerous historical anecdotes while tracing 20th century mathematical developments involved in these topics.)zbMATHGoogle Scholar

## Readings on Latin Squares

- Beck, A., Bleicher, M. N. and Crowe, D. W. (1972).
*Exccursions into Mathematics*, pp. 262–279. New York: Worth.Google Scholar - Crowe, D. W., and Thompson, T. M. (1987). Some modern uses of geometry. In
*Learning and Teaching Geometry*, iC-12, 1987*Yearbook*, M. M. Lindquist and A. P. Schulte (Eds.), pp. 101–112. Reston, VA: NCTM.Google Scholar - Gardner, M. (1959). Euler’s spoilers: The discovery of an order-10 Graeco-Latin square.
*Scientific American*201: 181–188.CrossRefGoogle Scholar - Sawyer, W. W. (1971). Finite arithmetics and geometries. In
*Prelude to Mathematics*, Chap. 13. New York: Penguin Books.Google Scholar