Summary Of Geometries

the book is an innovative modern exposition of Geometry, or rather, of geometries; it is the first textbook in which Felix kleins erlangen program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal-althoughh some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms œtoy geometries, the geometries of platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester Course lectures delivered at the independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing Upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean Geometry, albeit those lacking this knowledge may refer to a compendium in br>Chapter 0. Per the authors predilection, the book contains very little regarding the axiomatic approach to Geometry (save for a single br>Chapter on the history of non-euclidean Geometry), but two appendices provide a detailed treatment of Euclid and hilberts axiomatic. Perhaps the most important aspect of this course is the problems, which appear at the end of each br>Chapter, and are supplemented with answers at the conclusion of the text. By analysing and solving these problems, The reader will become capable of thinking and working geometrically, much more so than by simply learning the theory.

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