A Guide to Penrose Tilings / / by Francesco D'Andrea |
Autore | D'Andrea Francesco |
Edizione | [1st ed. 2023.] |
Pubbl/distr/stampa | Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 |
Descrizione fisica | 1 online resource (203 pages) |
Disciplina | 516.132 |
Soggetto topico |
Convex geometry
Discrete geometry Algebraic geometry Convex and Discrete Geometry Algebraic Geometry Mosaics (Matemàtica) |
Soggetto genere / forma | Llibres electrònics |
ISBN | 3-031-28428-3 |
Formato | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione | eng |
Nota di contenuto | Introduction -- Tilings and puzzles -- Robinson triangles -- Penrose tilings -- De Bruijn’s pentagrids -- The noncommutative space of Penrose tilings.-Some useful formulas. |
Record Nr. | UNINA-9910739449003321 |
D'Andrea Francesco | ||
Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023 | ||
Materiale a stampa | ||
Lo trovi qui: Univ. Federico II | ||
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Tilings of the plane : from Escher via Möbius to Penrose / / Ehrhard Behrends |
Autore | Behrends Ehrhard <1946-> |
Pubbl/distr/stampa | Wiesbaden, Germany : , : Springer, , [2022] |
Descrizione fisica | 1 online resource (284 pages) |
Disciplina | 516.132 |
Collana | Mathematics study resources |
Soggetto topico |
Tessellations (Mathematics)
Mosaics (Matemàtica) |
Soggetto genere / forma | Llibres electrònics |
ISBN |
9783658388102
9783658388096 |
Formato | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione | eng |
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. |
Record Nr. | UNINA-9910629301203321 |
Behrends Ehrhard <1946-> | ||
Wiesbaden, Germany : , : Springer, , [2022] | ||
Materiale a stampa | ||
Lo trovi qui: Univ. Federico II | ||
|
Tilings of the plane : from Escher via Möbius to Penrose / / Ehrhard Behrends |
Autore | Behrends Ehrhard <1946-> |
Pubbl/distr/stampa | Wiesbaden, Germany : , : Springer, , [2022] |
Descrizione fisica | 1 online resource (284 pages) |
Disciplina | 516.132 |
Collana | Mathematics study resources |
Soggetto topico |
Tessellations (Mathematics)
Mosaics (Matemàtica) |
Soggetto genere / forma | Llibres electrònics |
ISBN |
9783658388102
9783658388096 |
Formato | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione | eng |
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. |
Record Nr. | UNISA-996499870803316 |
Behrends Ehrhard <1946-> | ||
Wiesbaden, Germany : , : Springer, , [2022] | ||
Materiale a stampa | ||
Lo trovi qui: Univ. di Salerno | ||
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