José María Montesinos-Amilibia

Classical Tessellations and Three-Manifolds

• Broschiertes Buch

Produktbeschreibung

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.

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Produktdetails

• Universitext
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-540-15291-0
• 1987
• Seitenzahl: 256
• Erscheinungstermin: 1. September 1987
• Englisch
• Abmessung: 242mm x 170mm x 15mm
• Gewicht: 440g
• ISBN-13: 9783540152910
• ISBN-10: 3540152911
• Artikelnr.: 25647586

Inhaltsangabe

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.- 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 < 1.- 5.3 The manifolds of hyperbolic tessellations.- 5.4 The S1-action.- 5.5 Computing b.- 5.6 Involutions.- Appendix B.- The Hyperbolic Plane.- B.5 Metric.- B.6 The complex projective line.- B.7 The stereographic projection.- B.8 Interpreting G*.- B.10 The parabolic group.- B.11 The elliptic group.- B.12 The hyperbolic group.- Source of the ornaments placed at the end of the chapters.- References.- Further reading.- Notes to Plate I.- Notes to Plate II.Contents: S1-Bundles Over Surfaces.- Manifolds of Tessellations on the Euclidean Plane.- Appendix A: Orbifolds.- Manifolds of Spherical Tessellations.- Seifert Manifolds.- Manifolds of Hyperbolic Tessellations.- Appendix B: The Hyperbolic Plane.- Source of the Ornaments.- References.- Further Reading.- Notes to Plate I.- Notes to Plate II.