By A. B. Sossinsky

The publication is an cutting edge sleek exposition of geometry, or particularly, of geometries; it's the first textbook within which Felix Klein's Erlangen application (the motion of transformation teams) is systematically used because the foundation for outlining quite a few geometries. The process examine awarded is devoted to the proposition that each one geometries are created equal--although a few, in fact, stay extra equivalent than others. the writer concentrates on numerous of the extra special and lovely ones, which come with what he phrases "toy geometries", the geometries of Platonic our bodies, discrete geometries, and classical non-stop geometries. The textual content is predicated on first-year semester direction lectures introduced on the self sufficient college of Moscow in 2003 and 2006. it's in no way a proper algebraic or analytic remedy of geometric subject matters, yet fairly, a hugely visible exposition containing upwards of 2 hundred illustrations. The reader is anticipated to own a familiarity with straightforward Euclidean geometry, albeit these missing this data may possibly discuss with a compendium in bankruptcy zero. according to the author's predilection, the ebook includes little or no concerning the axiomatic method of geometry (save for a unmarried bankruptcy at the background of non-Euclidean geometry), yet Appendices offer an in depth remedy of Euclid's and Hilbert's axiomatics. probably an important element of this direction is the issues, which look on the finish of every bankruptcy and are supplemented with solutions on the end of the textual content. by way of reading and fixing those difficulties, the reader becomes in a position to considering and dealing geometrically, even more so than by means of easily studying the speculation. eventually, the writer makes the excellence among concrete mathematical items referred to as "geometries" and the singular "geometry", which he is familiar with as a fashion of pondering arithmetic. even if the ebook doesn't tackle branches of arithmetic and mathematical physics corresponding to Riemannian and Kähler manifolds or, say, differentiable manifolds and conformal box theories, the ideology of type language and transformation teams on which the booklet relies prepares the reader for the examine of, and finally, learn in those very important and speedily constructing components of latest arithmetic.