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.
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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.)
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Readings on Latin Squares
Beck, A., Bleicher, M. N. and Crowe, D. W. (1972). Exccursions into Mathematics, pp. 262–279. New York: Worth.
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.
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Sawyer, W. W. (1971). Finite arithmetics and geometries. In Prelude to Mathematics, Chap. 13. New York: Penguin Books.
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Cederberg, J.N. (2001). Axiomatic Systems and Finite Geometries. In: A Course in Modern Geometries. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3490-4_1
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DOI: https://doi.org/10.1007/978-1-4757-3490-4_1
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