Classical Tessellations and Three-Manifolds by Jose M. Montesinos-AmilibiaClassical Tessellations and Three-Manifolds by Jose M. Montesinos-Amilibia

Classical Tessellations and Three-Manifolds

byJose M. Montesinos-Amilibia

Paperback | October 1, 2009

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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.
Title:Classical Tessellations and Three-ManifoldsFormat:PaperbackDimensions:230 pages, 24.4 × 17 × 0.17 inPublished:October 1, 2009Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3540152911

ISBN - 13:9783540152910


Table of Contents

One.- S1-Bundles Over Surfaces.- 1.1 The spherical tangent bundle of the 2-sphere S2.- 1.2 The S1-bundles of oriented closed surfaces.- 1.3 The Euler number of ST(S2).- 1.4 The Euler number as a self-intersection number.- 1.5 The Hopf fibration.- 1.6 Description of non-orientable surfaces.- 1.7 S1-bundles over Nk.- 1.8 An illustrative example: IRP2 ? ?P2.- 1.9 The projective tangent S1-bundles.- Two.- Manifolds of Tessellations on the Euclidean Plane.- 2.1 The manifold of square-tilings.- 2.2 The isometries of the euclidean plane.- 2.3 Interpretation of the manifold of squaretilings.- 2.4 The subgroup ?.- 2.5 The quotient ?\E(2).- 2.6 The tessellations of the euclidean plane.- 2.7 The manifolds of euclidean tessellations.- 2.8 Involutions in the manifolds of euclidean tessellations.- 2.9 The fundamental groups of the manifolds of euclidean tessellations.- 2.10 Presentations of the fundamental groups of the manifolds M(?).- 2.11 The groups $$ \tilde \Gamma $$ as 3-dimensional crystallographic groups.- Appendix A.- Orbifolds.- A.1 Introduction. Table I.- A.2 Definition of orbifolds.- A.3 The 2-dimensional orbifolds, Table II.- A.4 The tangent bundle. Plates I and II.- Three.- Manifolds of Spherical Tessellations.- 3.1 The isometries of the 2-sphere.- 3.2 The fundamental group of SO(3).- 3.3 Review of quaternions.- 3.4 Right-helix turns.- 3.5 Left-helix turns.- 3.6 The universal cover of SO(4).- 3.7 The finite subgroups of SO(3).- 3.8 The finite subgroups of the quaternions.- 3.9 Description of the manifolds of tessellations.- 3.10 Prism manifolds.- 3.11 The octahedral space.- 3.12 The truncated-cube space.- 3.13 The dodecahedral space.- 3.14 Exercises on coverings.- 3.15 Involutions in the manifolds of spherical tessellations.- 3.16 The groups $$ \tilde \Gamma $$ as groups of tessellations of S3.- Four.- Seifert Manifolds.- 4.1 Definition.- 4.2 Invariants.- 4.3 Constructing the manifold from the invariants.- 4.4 Change of orientation and normalization.- 4.5 The manifolds of euclidean tessellations as Seifert manifolds.- 4.6 The manifolds of spherical tessellations as Seifert manifolds.- 4.7 Involutions on Seifert manifolds.- 4.8 Involutions on the manifolds of tessellations.- Five.- Manifolds of Hyperbolic Tessellations.- 5.1 The hyperbolic tessellations.- 5.2 The groups S?mn, 1/? + 1/m + 1/n <_20_1.-20_5.320_the20_manifolds20_of20_hyperbolic20_tessellations.-20_5.420_the20_s1-action.-20_5.520_computing20_b.-20_5.620_involutions.-20_appendix20_b.-20_the20_hyperbolic20_plane.-20_b.520_metric.-20_b.620_the20_complex20_projective20_line.-20_b.720_the20_stereographic20_projection.-20_b.820_interpreting20_g2a_.-20_b.1020_the20_parabolic20_group.-20_b.1120_the20_elliptic20_group.-20_b.1220_the20_hyperbolic20_group.-20_source20_of20_the20_ornaments20_placed20_at20_the20_end20_of20_the20_chapters.-20_references.-20_further20_reading.-20_notes20_to20_plate20_i.-20_notes20_to20_plate20_ii. 1.-="" 5.3="" the="" manifolds="" of="" hyperbolic="" tessellations.-="" 5.4="" s1-action.-="" 5.5="" computing="" b.-="" 5.6="" involutions.-="" appendix="" plane.-="" b.5="" metric.-="" b.6="" complex="" projective="" line.-="" b.7="" stereographic="" projection.-="" b.8="" interpreting="" _g2a_.-="" b.10="" parabolic="" group.-="" b.11="" elliptic="" b.12="" source="" ornaments="" placed="" at="" end="" chapters.-="" references.-="" further="" reading.-="" notes="" to="" plate="" i.-="">