Abstract
Finite geometries were developed in the late 19th 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.
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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. )
Anderson, I. (1974). A First Course in Combinatorial Mathematics. Oxford, England: Clarendon Press. (Chapter 6 discusses block designs and error-correcting codes.)
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.)
Benedicty, M., and Sledge, F.R. (1987). Discrete Mathematical Structures. Orlando, FL: Harcourt Brace Jovanovich. (Chapter 13 gives an elementary presentation of coding theory.)
Gensler, H.J. (1984). Gödel’s Theorem Simplified. Lanham, MD: University Press of America.
Hofstadter, D.R. (1984). Analogies and metaphors to explain Gödel’s theorem. In: D.M. Campbell and J.C. Higgins (Eds.), Mathematics: People, Problems, Results, Vol. 2, pp. 262–275. Belmont, CA: Wadsworth.
Kolata, G. (1982). Does Gödel’s theorem matter to mathematics? Science 218: 779–780.
Lockwood, J.R., and Runion, G.E. (1978). Deductive Systems: Finite and non-Euclidean Geometries. Reston, VA: N.C.T.M. (Chapter 1 contains an elementary discussion of axiomatic systems.)
Nagel, E., and Newman, J.R. (1956). Gödel’s proof. In: J.R. Newman (Ed.), The World of Mathematics, Vol. 3, pp. 1668–1695. New York: Simon and Schuster.
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. )
Smart, J.R. (1978). Modern Geometries,2nd ed. Belmont, CA: Wadsworth. (Chapter 1 contains an easily readable discussion of axiomatic systems and several finite geometries.)
Thompson, T.M. (1983). From Error-Correcting Codes Through Sphere Packings to Simple Groups. The Carus Mathematical Monographs, No. 21. Ithaca, NY: M.A.A. (Incorporates numerous historical antecdotes while tracing 20th century mathematical developments involved in these topics.)
Readings on Latin Squares
Beck, A., Bleicher, M.N., and Crowe, D.W. (1972). Excursions into Mathematics, pp. 262–279. New York: Worth.
Crowe, D.W., and Thompson, T.M. (1987). Some modern uses of geometry. In: M.M. Lindquist and A.P. Schulte (Eds.). Learning and Teaching Geometry, K-12, 1987 Yearbook, pp. 101-112. Reston, VA: N.C.T.M.
Gardner, M. (1959). Euler’s spoilers: The discovery of an order-10 Graeco-Latin square. Scientific American 201: 181–188.
Sawyer, W.W. (1971). Finite arithmetics and geometries. In: Prelude to Mathematics, Chap. 13. New York: Pengu in Books.
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© 1989 Springer Science+Business Media New York
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Cederberg, J.N. (1989). Axiomatic Systems and Finite Geometries. In: A Course in Modern Geometries. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3831-5_1
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DOI: https://doi.org/10.1007/978-1-4757-3831-5_1
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