By A. B. Sossinsky

The booklet is an leading edge smooth exposition of geometry, or quite, of geometries; it's the first textbook during which Felix Klein's Erlangen software (the motion of transformation teams) is systematically used because the foundation for outlining a number of geometries. The process research 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 unique 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 path lectures brought on the self sufficient college of Moscow in 2003 and 2006. it really is certainly not a proper algebraic or analytic therapy of geometric themes, yet quite, a hugely visible exposition containing upwards of two hundred illustrations. The reader is predicted to own a familiarity with straightforward Euclidean geometry, albeit these missing this information might seek advice from a compendium in bankruptcy zero. in keeping with the author's predilection, the e-book includes little or no in regards to the axiomatic method of geometry (save for a unmarried bankruptcy at the historical past of non-Euclidean geometry), yet Appendices supply a close therapy of Euclid's and Hilbert's axiomatics. might be an important point of this path is the issues, which seem on the finish of every bankruptcy and are supplemented with solutions on the end of the textual content. through interpreting and fixing those difficulties, the reader turns into in a position to pondering and dealing geometrically, even more so than via easily studying the speculation. finally, the writer makes the excellence among concrete mathematical items known as "geometries" and the singular "geometry", which he is familiar with as a fashion of considering arithmetic. even supposing the booklet doesn't deal with branches of arithmetic and mathematical physics akin to Riemannian and Kähler manifolds or, say, differentiable manifolds and conformal box theories, the ideology of type language and transformation teams on which the publication relies prepares the reader for the research of, and finally, study in those very important and swiftly constructing parts of up to date arithmetic.