Classical Tessellations and Three-Manifolds

§This unusual, richly illustrated book explores a relationship between classical tessellations and 3-manifolds. The space of positions of an euclidean, spherical or hyperbolic tessellation is a Seifert 3-manifold possessing one of three of Thurston's geometries. These manifolds are described in detail with an emphasis on their geometry. Classical tessellations for the spherical context are illustrated by colour photographs of minerals. A further set of colour plates depicts, for the first time, all 17 plane crystallographic groups in mosaics from the Alhambra of Granada (up till now it was thought only 13 could be found there). In his original and entertaining style and with numerous exercises and problems, the author introduces the reader to Seifert manifolds, classical tessellations, quaternions and rotations, orbifolds, 3-manifolds branched coverings etc. Graduate students will find in it a source of geometrical insight to low-dimensional topology. §Reserchers will find much that they already know clothed in a new garb - the framework of orbifolds - and will be able to use the text as a source of geometrical ideas for a low-dimensional topology seminar, for individual study projects for their students, or as the basis for a reading course. The account of spherical orbifolds will appeal to researchers in the foundations of crystallography. "Mas has dicho, Sancho, de 10 que sabes (dixo Don Quixote), que hay algunos que se cansan en saber, y averiguar cosas que despues de sabidas, y averiguadas, no importa un ardite al entendimiento, ni a la memoria. " "You have said more than you know, Sancho", said Don Quixote, "for there are some who tire them selves out learning and proving things which, once learnt and proved, do not concern either 'the under standing 01' the memory a jot. " Cervantes, Don Quixote, Part II, Chapter LXXV, Of the great Adventure of Montesinos' Cave in the heart of La Mancha, which the valorous Don Quixote brought to a happy ending. This book explores a relationship between classical tessellations and three-manifolds. All of us are very familiar with the symmetrical ornamental motifs used in the decoration of walls and ceilings. Oriental palaces contain an abundance of these, and many examples taken from them will be found in the following pages. These are the so-called mosaics or symmetrical tessellations of the euclidean plane. Even though we can imagine or even create very many of them, in fact the rules governing them are quite restrictive, if our purpose is to understand the symmetric group of the tessellation, that is to say, the group consisting of the plane isometries which leave the tessel lation invariant.