Classical Tesselations and Three-Manifolds

"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 [...]
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"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.