# Geometric Symmetry in Patterns and Tilings by C E Horne

Geometric Symmetry in Patterns and Tilings
by C E Horne
Contents
Foreword ix
Acknowledgements x
1 Introduction 1
References 4
2 Classification of designs by symmetry group 7
2.1 Introduction 7
2.2 Symmetry and its relevance to designs 7
2.3 Symmetry operations 7
2.4 Symmetry group 10
2.5 Figures and designs 10
2.6 Classification of finite designs 11
2.7 Structure of translational designs 14
2.8 Generating functions 25
2.9 Classification of monotranslational designs 28
2.10 Classification of ditranslational designs 29
2.11 Construction of finite designs 33
2.12 Construction of monotranslational designs 39
2.13 Construction of ditranslational designs 47
2.14 Summary 76
References 77
3 Classification of designs by symmetry group and design unit 79
3.1 Introduction 79
3.2 Notation 82
3.3 Finite designs 82
3.4 Monotranslational designs 82
3.5 Ditranslational designs 82
3.6 Construction of finite designs 94
3.7 Construction of monotranslational designs 98
3.8 Construction of ditranslational designs 110
3.9 Summary 126
References 129
4 Classification of discrete patterns 131
4.1 Introduction 131
4.2 Monomotif pattern 131
4.3 Discrete pattern 133
4.4 Primitive pattern 135
4.5 Induced motif groups 137
4.6 Motif-transitive subgroups 142
4.7 Classification of finite discrete pattern types 145
4.8 Classification of monotranslational discrete pattern types 149
4.9 Classification of ditranslational discrete pattern types 149
4.10 Construction of finite pattern types 151
4.11 Construction of monotranslational discrete pattern types 151
4.12 Construction of ditranslational discrete pattern types 158
4.13 Summary 170
References 170
5 Classification of isohedral tilings 171
5.1 Introduction 171
5.2 Isohedral tiling 172
5.3 Dirichlet tiling 178
5.4 Topology of tilings 180
5.5 Incidence symbols 189
5.6 Marked isohedral tilings 196
5.7 Classification of finite isohedral tiling types 197
5.8 Classification of monotranslational isohedral tiling types 197
5.9 Classification of ditranslational isohedral tiling types 203
5.10 Construction of finite isohedral tiling types 203
5.11 Construction of monotranslational isohedral tiling types 205
5.12 Construction of ditranslational isohedral tiling types 211
5.13 Summary 230
References 232
6 Summary and conclusions 233
References 236
Index 237

Foreword
Geometric Symmetry in Patterns and Tilings results from one of a series of exciting and innovative research projects emanating from the School of Textile Industries at University of Leeds.

This particular project was conducted under my supervision, and was aided by scholarship funding from the Worshipful Company of Clothworkers of the City of London. It extends the Leeds tradition of research into pattern symmetry initiated in the 1930s by H J Woods, a physicist (and mathematician), whose contribution in laying the foundations for current thinking on the geometrical characteristics of patterns is, today, widely acknowledged by scholars in the field.

Whilst many symmetry concepts have their origin in the area of crystallography, an appreciation of their usefulness has, in recent years, extended to many disciplines and realms of study.Washburn and Crowe made a major contribution in the area of anthropology in their largely pioneering work Symmetries of Culture. The mathematical treatise Tilings and Patterns by Grünbaum and Shephard stands as a major contribution to the conceptual development of the subject. Visions of Symmetry, Schattschneider’s monumental study of the work of M C Escher, has not only stimulated an insight into the periodic drawings and patterns of the artist but has also encouraged an understanding of symmetry concepts beyond a mathematically aware audience to inspire the creation of original decorative patterns.

Recent research projects at Leeds have employed symmetry concepts in the investigation of patterns produced in a range of historical and/or cultural contexts and as a systematic means of generating printed-textile designs. Layer symmetry principles have been employed in the analysis of woven-fabric structures, and as a basis for developing a systematic means of designing woven fabrics. The present book focuses principally on characteristics of surface-pattern design, and presents a comprehensive means of classifying patterns and tilings. A wide range of original illustrative material is included.

M A Hann,
University of Leeds

Acknowledgements
This book has been developed from research activities undertaken whilst studying in the School of Textile Industries at the University of Leeds. Consequently, first I would like to express my gratitude to the School of Textile Industries, the Worshipful Company of Clothworkers, and in particular to my supervisor, Dr. M A Hann, and the Head of Department at the time, Professor D Johnson, for supporting my research.

I am also sincerely grateful to all my family and friends for their support, encouragement and understanding whilst I have been compiling this work. I would especially like to thank my family, Brenda, Tony, Christopher, Alison and Jenny. I am also greatly indebted to many friends who have shown their continual care and consideration, in particular Rachel Segal, Mark Colpus, Graham Gifford, Marion Small and Wendy Cawthray.

Finally, I would like to thank Woodhead Publishing for maintaining their interest in my work and particularly Patricia Morrison for showing such patience and support over a long period of time.