Table Of ContentMathematics 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 mathe-
matics 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
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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 prin-
ciple 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 archi-
tecture (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.
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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 architec-
tural composition founded on the algebraic structure of symmetry groups in mathe-
matics 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.
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