Magic Polygons (Raymond R. Fletcher Ill)


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The concept of a magic square is of ancient origin; the simple 3 by 3 square called “Lo Shu” being the first to appear. The cells of a magic square are populated with consecutive positive integers in such a way that all rows and columns have the same sum. “Magic Polygons” presents a natural generalization which involves polygons with more than 4 sides. A polygon is first tiled with parallelograms with the result that sequences of parallelograms with mutually parallel sides form streams which extend from one boundary edge to an opposite boundary edge. These streams form the analog of rows and columns in a magic square: the tiles are labeled with integers so that all streams have the same sum. Instead of using the positive integers 1, 2, 3, … to label the tiles, various abelian groups are used, thus adding a further new dimension to the constructions. Related topics such as “Latin Polygons”, “Magic Cylinders” and “Magic Circle Systems” are also introduced.


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