Islamic Geometric Patterns Jay Bonner Islamic Geometric Patterns Their Historical Development and Traditional Methods of Construction with a chapter on the use of computer algorithms to generate Islamic geometric patterns by Craig Kaplan Jay Bonner Bonner Design Consultancy Santa Fe, New Mexico, USA With contributions by Craig Kaplan University of Waterloo Waterloo, Ontario, Canada ISBN 978-1-4419-0216-0 ISBN 978-1-4419-0217-7 (eBook) DOI 10.1007/978-1-4419-0217-7 Library of Congress Control Number: 2017936979 # Jay Bonner 2017 Chapter 4 is published with kind permission of # Craig Kaplan 2017. All Rights Reserved. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover photograph: # The Trustees of the Chester Beatty Library, Dublin: CBL Is 1431, ff. 7b-8a Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media LLC The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A. For my wife, Shireen, without whose support this book would never have seen the light of day. Jay Bonner For Reza Sarhangi, who convinced me to cross a bridge and never look back. Craig Kaplan Foreword For over eleven centuries, Islamic artists and architectural designers have developed extraor- dinary skills to produce many amazing forms of decoration. This superbly illustrated book, by Jay Bonner (with a final chapter by Craig Kaplan), is a wonderfully comprehensive and deeply thoughtful account of these highly distinctive designs. I feel sure that this book will represent a landmark in the study of Islamic design. These remarkable artworks take various forms, but they are almost always of an entirely nonrepresentational character, only a very few showing indications of recognizable natural objects, such as leaves or flowers. Yet there is a distinctive beauty in these designs, most of these patterns being of a particular geometrical character, demonstrating a keen and subtle knowledge and interest in geometry and a profound skill in using geometrical motifs to produce some incredibly intricate patterns. These would normally be repetitive, in a planar arrangement, though sometimes bent to cover a curved surface, such as for a dome or ceiling. They gain some of their beauty from the intricacy and ingenuity of the designs, which frequently involve sophisticated geometrical ideas in unexpected ways. These artists had clearly developed some significant understanding of mathematical notions that were not properly developed by professional mathematicians until the early twentieth century, when the plane Euclidean symmetry groups (“wallpaper symmetries”) were finally classified into 17 distinct types. Representations of these different symmetries are copiously exhibited in Islamic patterns and, remarkably, all 17 are to be found in the single location of the Alhambra Palace in Granada, Spain, constructed more than half a millennium earlier than the time that these different symmetries were explicitly distinguished by mathematicians. In the early 1930s, these Alhambra designs had a seminal inspirational influence on the extraordinary work of the well-known Dutch artist M.C. Escher. Yet, there is much more than the illustration of different Euclidean symmetries in the ingenious and intricate use of geometry in these Islamic designs. There is no doubt about the skill and artistry that have gone into the production of these creations. There is much, also, which is of mathematical interest in the local symmetries that have been frequently incorporated in this way, where we find symmetrical star-shaped regions of all sorts of improbable-seeming symmetry, such as combinations of stellar shapes with 13-fold and 9-fold symmetry, which are quite ruled out as overall symmetries of crystal-like arrangements, for which only 2-fold, 3-fold, 4-fold, and 6-fold symmetries are allowed. The question has sometimes been raised, as to whether these ancient Islamic geometers might have discovered and made use of quasisymmetrical features, such as the fivefold ones that I found myself in the mid-1970s. These would give rise to patterns that “almost” (in a technical sense) repeat, but which never quite do so, and the fivefold symmetry is likewise a feature that “almost” holds but not quite. When I found such designs that arise naturally as the only arrangements in which one can assemble a pair of “jig-saw” shapes, I was particularly struck by the simplicity of these particular shapes, which could indeed force such fivefold quasisymmetric patterns. I thought it not unlikely that things of this nature might well have been made use of by the Islamic geometric artists of antiquity. Indeed, various later researchers have claimed that there is indeed evidence that some ancient Islamic buildings vii viii Foreword might well employ such quasisymmetric features. In this book, Jay Bonner argues otherwise, finding strong indications that the desire for exact periodicity was an overriding driving motive for the Islamic patterns, and that there is no evidence that quasisymmetric features were ever made use of. It is, of course, still possible that convincing evidence might eventually come to light that the ideas of quasisymmetry did play a role for some of the ancient Islamic designers. However, I am inclined to agree with Jay Bonner that strict periodicity seems to have played such a key role for these ancient geometric artists that quasisymmetric considerations would be unlikely. It is, to me, quite extraordinary how some of these Islamic designers were able to fit such improbable symmetries as 13-fold symmetric stars combined with 9-fold ones so as to create a design with such a natural-looking elegance, and a periodicity that appears at a distance not much greater than the extent of these stellar shapes themselves. Mathematical quasisymmetric patterns with, say 11-fold or 13-fold quasisymmetry, can now be produced in computer- generated pictures, but it would only be at extremely remote places where local regions with this symmetry would be evident. If one is looking for beauty of design, with such regions of, say, 11-fold or 13-fold symmetry, then the ancient Islamic designs win hands down! Roger Penrose Emeritus Rouse Ball Professor of Mathematics Mathematical Institute University of Oxford Preface and Acknowledgements More than with any other cultural heritage, the work of Muslim artists over the centuries reveals the visual beauty that is inherent to geometry. Drawing inspiration from geometry led to an abundance of aesthetic innovations within the tradition of Islamic geometric design that were inexorably associated with methodological practices. Of particular importance was the discovery that geometric patterns could be extracted from underlying polygonal tessellations, herein referred to as the polygonal technique. Over time, and in the hands of skilled and dedicated practitioners, this design methodology engendered the extraordinary breadth of diversity that characterizes this artistic tradition. As architectural ornament, Quranic illumi- nation, and its more limited use within the applied arts, geometric designs were enthusiasti- cally embraced by succeeding Muslim cultures, and along with calligraphy and the floral idiom, became an integral aspect of Islamic aesthetics. As such, its role within Islamic art as a whole is paramount. Yet scholarship of this vast subject remains underrepresented in two key areas: historical development and design methodology. Related to this relative lack of attention is the surprising absence of a comprehensive approach to categorizing the range of patterns within this tradition. Additionally, while the exceptional achievements of past geometric artists serve as a source of inspiration for many contemporary artists, designers, craftspeople, and architects, the loss of methodological knowledge associated with this artistic tradition has substantially thwarted those with an interest in incorporating such designs into their work. In consideration of these gaps this book has several intentions: to provide a greater understanding of the developmental history of this remarkable artistic tradition; to emphasize a more nuanced attribution of the geometric and design diversity that is a hallmark of this distinctive form of art; and to provide a detailed elucidation of the methodological practices that are responsible