Tilings of the plane : from Escher via Möbius to Penrose / / Ehrhard Behrends

• Record Nr.: UNISA996499870803316
• Autore: Behrends Ehrhard <1946->
• Pubbl/distr/stampa: Wiesbaden, Germany : , : Springer, , [2022]; ©2022
• ISBN: 9783658388102; 9783658388096
• Descrizione fisica: 1 online resource (284 pages)
• Collana: Mathematics study resources ; ; Volume 2
• Disciplina: 516.132
• Soggetti: Tessellations (Mathematics); Mosaics (Matemàtica); Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- Part I Looking over Escher's Shoulder -- 2 Symmetries and Fundamental Domains -- 2.1 What is Symmetry? -- 2.2 What Movements are There? -- 2.3 Groups of Movements -- 2.4 Discontinuous Groups and Fundamental Domains -- 3 The Discontinuous Symmetry Groups of the Plane -- 3.1 How Many Different Groups of Movements are There? -- 3.2 Finite Groups of Movements -- 3.3 The Subgroup of Translations -- 3.4 The 7 Frieze Groups -- 3.4.1 : Only Translations -- 3.4.2 : Only Reflections of Type 1 () -- 3.4.3 : Only Reflections of Type 2 () -- 3.4.4 : Proper Glide Reflections () -- 3.4.5 : Only Rotations () -- 3.4.6 : Rotations, Type-1 and Type-2 Reflections () -- 3.4.7 : Proper Glide Reflections, Type-2 Reflections, and Rotations () -- 3.4.8 Summary -- 3.4.9 Classification: A Test -- 3.4.10 Hints for Artists -- 3.5 The 17 Plane Crystal Groups -- 3.5.1 The Crystallographic Restriction -- 3.5.2 Translations, Reflections: 4 Groups -- 3.5.3 Translations, 2-Rotations, Reflections: 5 Groups -- 3.5.4 Translations, 3-Rotations, (Glide) Reflections: 3 Groups -- 3.5.5 Translations, 4-Rotations, Reflections: 3 Groups -- 3.5.6 Translations, 6-Rotations, Reflections: 2 Groups -- 3.5.7 Classification: A Test -- 4 The Heesch Constructions -- 4.1 Lattices ; and Nets -- 4.2 The Heesch Construction: Motivation -- 4.3 The Heesch Constructions: 28 Methods -- References for Part I -- Part II Möbius Transformations -- 5 Möbius Transformations -- 5.1 Complex Numbers: Some Reminders -- 5.2 Möbius Transformations: Definitions and First Results -- 5.3 Möbius Transformations and Circles -- 5.4 Fixed Points of Möbius Transformations -- 5.5 Conjugate Möbius Transformations -- 5.6 Characterization: Fixed Points in {0,∞} -- 5.7 Characterization: the General Case -- 5.8 Wish List/Visualization -- 6 Groups of Möbius Transformations.; 6.1 First Examples of Groups of Möbius Transformations -- 6.2 Fundamental Domains and Discrete Groups -- 6.3 Special Möbius Transformations -- 6.4 Digression: Hyperbolic Geometry -- 6.4.1 Hyperbolic Geometry I: The Upper Half-plane -- 6.4.2 Hyperbolic Geometry II: The Unit Circle -- 6.5 The Modular Group -- 6.6 Groups with Two Generators -- 6.7 Schottky Groups -- 6.8 The Mystery of the Parabolic Commutator -- 6.9 The Structure of Kleinian Groups -- 6.9.1 The Isometric Circles -- 6.9.2 The Limit Set -- 6.9.3 A Fundamental Domain -- 6.10 Parabolic Commutators: Construction -- References for Part II -- Part III Penrose Tilings -- 7 Penrose Tilings -- 7.1 Non-periodic Tilings: The Problem -- 7.2 The "Golden" Penrose Triangles -- 7.3 Which Tiling Patterns are Possible? -- 7.4 Index Sequences Generate Tilings -- 7.5 Isomorphisms of Penrose Tilings -- 7.6 Supplements -- References for Part III -- Index.