finite geometry

plural finite geometries

1. countableuncountablegeometryAny geometric system that has only a finite number of points.

   Finite geometries, such as Euclidean and projective geometries, are powerful mathematical tools for constructing error-control codes.

   2009, William Ryan, Shu Lin, Channel Codes: Classical and Modern, Cambridge University Press, page 430

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   But the work of von Staudt, Fano, and Veblen led even to the construction of finite geometries having only finitely many points and lines. Originally having only incidence relations, these finite geometries have been extensively developed through the introduction of suitable axioms of order and congruence by Järnefelt and Kustaanheimo, who have proposed their use in physics to solve certain paradoxes due apparently to the use of continuous variables.

   1980, Judson Chambers Webb, Mechanism, Mentalism and Metamathematics: An Essay on Finitism, Springer, page 39

   A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed.

   1999, S. Ball, “Polynomials in Finite Geometries”, in J. D. Lamb, D. A. Preece, editors, Surveys in Combinatorics, 1999, Cambridge University Press, page 17

2. uncountablegeometryThe branch of geometry that concerns geometric systems with only a finite number of points.

   The coordinates of geometry were as incompatible with representation theory as were the splitting fields of representation theory with finite geometry.

   1983, Robert A. Liebler, “Combinatorial Representation Theory and Translation Planes”, in Norman L. Johnson, Michael J. Kallaher, Calvin T. Long, editors, Finite Geometries: Proceedings of a Conference, CRC Press, page 307

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   We give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets.

   1996, Dieter Jungnickel, “Maximal Sets of Mutually Orthogonal Latin Squares”, in S. Cohen, H. Niederreiter, editors, Finite Fields and Applications: Proceedings of the 3rd International Conference, Cambridge University Press, page 129

   Finite geometry is a broad area of research, but much of it is not Euclidean in flavour and no theory seems to match that of oriented matroids in establishing combinatorial Euclidean geometry.

   2007, Michael B. Smyth, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian E. Pratt-Hartmann, Johan F. A. K. van Benthem, editors, Handbook of Spatial Logics, Springer, page 787