This book is an introduction to the geometry of Euclidean, affine, and projective spaces with special emphasis on the important groups of symmetries of these spaces. The two major objectives of the text are to introduce the main ideas of affine and projective spaces and to develop facility in handling transformations and groups of transformations. Since there are many good texts on affine and projective planes, the author has concentrated on the n-dimensional cases.Designed to be used in advanced undergraduate mathematics or physics courses, the book focuses on “practical geometry,” emphasizing topics and techniques of maximal use in all areas of mathematics. These topics include:Algebraic and Combinatoric PreliminariesIsometries and SimilaritiesAn Introduction to CrystallographyFields and Vector SpacesAffine SpacesProjective SpacesSpecial features include a spiral approach to symmetry; a review of the algebraic prerequisites; proofs which do not appear in other texts, such as the Polya-Burnside theorem; an extensive bibliography; and a large collection of exercises together with suggestions for term-paper topics. In addition, special emphasis is placed on the geometric significance of cosets and conjugates in a group.