The aim of the book is to study symmetries and tesselation, which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process, the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book.
In Part I, it is shown that there are 17 principally different possibilities of tesselation of the plane, the so-called "plane crystal groups". Complementary to this, ideas of Harald Heesch are described, who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself - following in the footsteps of Escher, so to speak - to create artistically sophisticated tesselation.
In the corresponding investigations for the complex plane in Part II, movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Möbius transformations: Kleinian groups, Schottky groups, etc. There are also interesting connections to hyperbolic geometry.
Finally, in Part III, a third aspect of the subject is treated, the Penrose tesselation. This concerns results from the seventies, when easily describable and provably non-periodic parquetisations of the plane were given for the first time.
Author(s): Ehrhard Behrends
Series: Mathematics Study Resources, 2
Edition: 1st ed. 2022
Publisher: Springer
Year: 2022
Language: English
Pages: 283
City: Wiesbaden
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
