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Mathematics and the Built Environment 6 Jin-Ho Park Architectural and Urban Subsymmetries Mathematics and the Built Environment Volume 6 Series Editors Kim Williams, Kim Williams Books, Torino, Italy Michael Ostwald , Built Environment, University of New South Wales, Sydney, NSW, Australia Throughout history a rich and complex relationship has developed between mathematics and the various disciplines that design, analyse, construct and maintain the built environment. This book series seeks to highlight the multifaceted connections between the disciplines of mathematics and architecture, through the publication of monographs that develop classical and contemporary mathematical themes – geometry, algebra, calculation, modelling. These themes may be expanded in architecture of any era, culture or style, from Ancient Greek and Rome, through the Renaissance and Baroque, to Modernism and computational and parametric design. Selected aspects of urban design, architectural conservation and engineering design that are relevant for architecture may also be included in the series. Regardless of whether books in this series are focused on specific architectural or mathematical themes, the intention is to support detailed and rigorous explorations of the history, theory and design of the mathematical aspects of built environment. Jin-Ho Park Architectural and Urban Subsymmetries Jin-Ho Park Department of Architecture Inha University Incheon, Korea (Republic of) ISSN 2512-157X ISSN 2512-1561 (electronic) Mathematics and the Built Environment ISBN 978-3-031-08945-9 ISBN 978-3-031-08946-6 (eBook) https://doi.org/10.1007/978-3-031-08946-6 Mathematics Subject Classification: 00A67, 05E05, 06A06, 20B30, 97M80 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For my wife, Juhee Cho and son, Yoojun Park Preface Designs in art and architecture are frequently encountered in a random or ordered manner. Although they are composed of the same constituents, some designs are characterized by a completely disordered organization of their constituents at first glance. By contrast, some designs exhibit a straightforward order. Their differences may rely on the underlying structure of spatial logic. Similarly, although some objects in nature are composed of the same constituent elements, they appear to be completely different. For example, diamonds are composed of a single element, i.e., carbon. Diamonds are regarded as precious gemstones because of their beauty and quality, not because they are made of carbon, but because of the way carbon is organized. The underlying structure can be compared with that of graphite. Although distinct constituents in a design are combined or interconnected, the underlying structure of their spatial order may be discussed regarding the use of symmetry principles in mathematics. Architects have adopted a mathematical principle in architectural design due to its logical, explicit, and explainable nature. With such principle, architects can animatedly engage in architectural practices. Throughout the history of architecture, symmetry has served as a crucial notion that provides insight into the balance and harmony of spatial composition in architecture (Vitruvius 1931; Williams 1999). The term, which is derived from the Latin word symmetria (a portmanteau of syn [together] and metria [to measure]), indicates the appropriate arrangement of the parts of a whole to one another regarding size and form. It clearly implies compositional balance between parts to create a harmonious whole. Studying the symmetry principles of spatial forms may facilitate the understanding and development of architectural designs. These principles help reveal different hier- archical levels wherein various types of symmetry or subsymmetry are superimposed in individual designs. These principles illustrate how symmetry may be used deliber- ately in the design process. Although used directly, the interaction between local and global subsymmetries is of particular interest. Subsymmetries may be contested in a single design. Then, the overall design becomes astonishingly complex and asym- metric in the end. Thus, the use of symmetry may sometimes be immediately apparent by simply looking into designs, although the final design is seemingly asymmetrical. vii viii Preface Various symmetries or subsymmetries may also be manifested in parts of a design but not immediately recognizable despite a nearly obsessive concern for symmetry. The advantage of operating with symmetry concepts in this manner is to provide architects and designers with an explicit method for understanding the symmetrical logics of sophisticated designs and gaining insights into the construction of new designs by using symmetry operations. The current study is a crucial prerequisite for increasing the understanding of complex designs and the creation of new architectural forms (Weyl 1952). Many ideas on the principles of form have been introduced. For example, Goethe’s theory of metamorphosis in The Metamorphosis of Plants (Goethe 1946) focuses on the notion that an ideal form or urform exists. It is the vital key to understanding the subsequent development of forms. In the nineteenth century, a German architect and theorist, Gottfried Semper borrowed the term “tectonics” to delineate the theoretical opinion of the common principles of forms in nature and in art, writing, “[t]ectonics is an art that takes nature as a model—not nature’s concrete phenomena but the uniformity [Gesetzlichkeit] and the rules by which she exists and creates” (Hermann 1989). For him, the understanding of basic principles would aid the creation of new work through modification and transformation. Louis Sullivan in his book entitled A System of Architectural Ornament (1924) promoted the notion of “the Seed-Germ” as the fundamental principle from which a plant will grow. It is the fundamental notion of biological growth from which several different designs can be evolved through geometric methods. Following Sullivian’s notion of organic form, Frank Lloyd Wright, in his article “In the Cause of Architecture: Composition as Method in Creation”, even laid emphasis on the study of the principles of composition, claiming that “geometry [is] at the center of every Nature-form we see” (Wright 1928). For those above-mentioned, the fundamental principles of forms in nature and art play a significant role in understanding and producing a variety of designs. The paramount significance of the idea is similarly found in other creative disciplines, such as Arnold Schoenberg’s notable 12-tone technique for musical composition in music (Boss 2014) and Johannes Itten’s color circle used in the basic course at the Bauhaus (Itten 1970). The tradition continues to positively influence creativity. This book has two objectives: to explore the fundamental principles of architectural composition founded on the algebraic structure of symmetry groups in mathematics and to apply the principles in the analysis and synthesis of architectural and urban designs. Thus, this book is logically divided into two parts. Part I contains the fundamental principles and techniques derived from the mathematical principles of symmetry, groups, subgroups, and isometries. Part II presents the composition and decomposition of architecture and urban designs. The complementary approach in the dialectic between analysis and synthesis is subtle in such a study. Analytically, by viewing and decomposing architectural and urban designs in this manner, different subsymmetries superimposed in several layers in a design that may not be immediately recognizable become transparent. Synthetically, architects can benefit from being conscious of using group operations and spatial transformations associated with symmetry in compositional and thematic Preface ix development. Architects may sometimes discover emergent designs that they have never anticipated in the beginning of the design process. Hence, the fundamental principles in Part I will give the readers theoretical and methodical background to the clearer understanding of complex existing designs. Also, these principles act as a source of knowledge for new design development. Hence, rigorous research of historic precedents is extremely valuable for notable design progress. Then, this book shows some potential uses of the principles in the creation of new composition in architecture. Therefore, much of Part II is used to provide a reference for further study. Incheon, Korea (Republic of) Jin-Ho Park Acknowledgments This book is deeply indebted to the late Lionel March, in whose series of courses, “Fundamentals of Architectonics,” offered at University of California, Los Angeles. It was from the courses and discussions that the idea of writing about this book emerged. I also thank George Stiny and Terry Knight for their constant supports and scholarly influences. In terms of the actual production of the book, I would like to acknowledge my debt to the researchers and staff of the Design Research and Innovation Lab at Inha University, Korea for their constant and invaluable help during the seemingly endless period of research and preparation that that preceded the publication of this text, particularly, Sujung Ji and Sejung Jung for their fine effort in preparing some of the drawings for the book. Among recent graduate students, I thank Donggyun Ro, Hakju Lee, Reem Abdelkader, Seungbeom Park, Minwook Mo, Youjin Park, Myungsoo Kim, Nguyen Vu Tuan Anh, Yangsook Jeon, Eunsun Jung, Sangha Lee, and Jihyoung Cho for their various contributions. There are a number of other scholars and professors who indirectly have played a key role in the development of this book. Among these, particular credit should be accorded to Athanassios Economou, Dirk Huylebrouck, Denes Nagy, Beatrix Mecsi, Ji-Hyun Lee (KAIST), and Thi Hong Na Le for their consistent comments and reviews. Also, professor Jiyoung Park, Minjung Cho, Seung Wan Hong, and Hyejin Jung have been appreciated colleagues in the Department of Architecture at Inha University. It should go without saying that there are countless others to whom I am also equally indebted and whose specific contributions are acknowledged. Last, but not least, I especially appreciate the recommendation and encouragement of book series editors, Kim Williams and Michale J. Ostwald. Without their hard and timely work and focused attention to producing an impeccable product, it would never come together as quickly as it did. Especially, I am deeply gratitude to Kim Williams for her sincere comments to my early studies of subsymmetry that were published in the Nexus Network Journal. xi