Mirrors and Reflections Mirrors and Reflections presents an intuitive and elementary introduction to finite reflection groups. Starting with basic principles this book provides a comprehensive classification of the various types of finite reflection groups and describes their underlying geometric properties. Unique to this text is its emphasis on the intuitive geometric aspects of the theory of reflection groups making the subject more accessible to the novice. Primarily self contained necessary geometric concepts are introduced and explained. Principally designed for coursework this book is saturated with exercises and examples of varying degrees of difficulty. An appendix offers hints for solving the most difficult problems. Wherever possible concepts are presented with pictures and diagrams intentionally drawn for easy reproduction. Finite reflection groups is a topic of great interest to many pure and applied mathematicians. Often considered a cornerstone of modern algebra and geometry an understanding of finite reflection groups is of great value to students of pure or applied mathematics. Requiring only a modest knowledge of linear algebra and group theory this book is intended for teachers and students of mathematics at the advanced undergraduate and graduate levels. Geometric Background Affine Euclidean Space Isometries of Hyperplane Arrangements Polyhedral Cones Mirrors Reflections Roots Mirrors and Reflections Systems of Mirrors Dihedral Groups Root Systems Root Systems An1 BCn Dn Coxeter Complexes Chambers Generation Coxeter Complex Residues Generalized Permutahedra Classification Generators and Relations Classification of Finite Reflection Groups Construction of Root Systems Orders of Reflection Groups Three Dimensional Reflection Groups Reflection Groups in Three Dimensions Icosahedron.