Tilings of the Plane: From Escher via Möbius to Penrose

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