Distinguishing geometry as a way of thinking about shapes from a geometry as a set with a transformation group acting on it, Sosinskii (mathematics, Independent U. of Moscow) describes some of the latter that he finds most elegant. Topics include Euclidean geometry, toy geometries and main definitions, discrete subgroups of the isometry group of the plane and tilings, the Poincaré half-plane model, hyperbolic trigonometry and absolute constants, and the hierarchy of geometries. A Russian language edition was published by the Moscow Center of Continuous Mathematical Education in 2006 and is used at Independent University. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)