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Description
Geometric enumeration treats of the enumeration of geometrical forms in geometrical patterns. A simple example is the determination of how many squares in a checkerboard, taking into account the squares of various sizes. The literature on geometrical enumeration is quite meager being almost entirely restricted to the treatment of a few specific problems in various journals and books. These problems are usually narrow in scope, such as the example given above. It is the purpose of this paper to fill this gap in the mathematical literature by providing a unified treatment of geometrical enumeration encompassing the various analytic methods which may be employed and by providing the solution of a number of problems in this field. In the treatment of problems an effort will be made to treat each problem in its most general sense. The aforementioned checkerboard problem is an example of a specific problem in a restricted sense. The same problem in its most general sense would be the question of how many squares in an m by m checkerboard. As will be seen, even this problem will appear as part of a more general problem. The treatment begins with the study of a simple triangular pattern showing various methods of attacking the problem. The method ultimately developed is the method of deriving summation formulas and algebraic reduction thereof. As this method is regarded as the most efficient for most problems in geometric enumeration, it is the method which is employed in subsequent analyses, wherever it is suitably applicable. The subsequent chapters begin with a treatment of certain extensions of a simple triangular pattern including an application to the Chinese checkerboard. A detailed study is then made of a complex triangular pattern illustrating the broad scope of the methods employed. The foregoing problems involve only one basic parameter in the development of the pattern. A two parameter pattern is tested in the study of a rectangular design. Finally, an analysis of triangles in convex polygons is provided. The analyses in this paper are, in general, confined to the place except, in one instance, where the extension of a particular pattern to space is readily made. However, the methods developed in this paper can usually be employed for various types of spatial patterns.
