FROM NUMBERS TO DIGITS: ON THE CHANGING ROLE OF MATHEMATICS IN ARCHITECTURE A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY BETÜL KOÇ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARCHITECTURE IN ARCHITECTURE JUNE 2008 Approval of the thesis: FROM NUMBERS TO DIGITS: ON THE CHANGING ROLE OF MATHEMATICS IN ARCHITECTURE submitted by BETÜL KOÇ in partial fulfillment of the requirements for the degree of Master of Architecture in Department of Architecture, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences __________________ Assoc. Prof. Dr. Güven Arif Sargın Head of Department, Architecture __________________ Assoc. Prof. Dr. Zeynep Mennan Supervisor, Department of Architecture, METU __________________ Examining Committee Members Assoc. Prof. Dr. Selahattin Önür Department of Architecture, METU ___________________ Assoc. Prof. Dr. Zeynep Mennan Department of Architecture, METU ___________________ Inst. Dr. Namık Erkal Department of Architecture, METU ___________________ Inst. Dr. Haluk Zelef Department of Architecture, METU ___________________ Assoc. Prof. Dr. Arda İnceo ğlu Department of Architecture, İTU ___________________ Date: June 20, 2008 ii I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Betül Koç iii ABSTRACT FROM NUMBERS TO DIGITS: ON THE CHANGING ROLE OF MATHEMATICS IN ARCHITECTURE Koç, Betül M. Arch., Department of Architecture Supervisor: Assoc. Prof. Dr. Zeynep Mennan June 2008, 156 pages This study is a critical reconsideration of architecture’s affiliation with mathematics and geometry both as practical instrument and theoretical reference. The thesis claims that mathematics and its methodological structure provided architects with an ultimate foundation and a strong reference outside architecture itself ever since the initial formations of architectural discourse. However, the definitive assumptions and epistemological consequences of this grounding in mathematical clarity, methodological certainty and instrumental precision gain a new insight with the introduction of digital technologies. Since digital technologies offer a new formation for this affiliation either with their claim of a better geometric representation or mathematical controllability of physical reality (space), the specific focus on these newly emerging technologies will be developed within a theoretical frame presenting the significant points of mathematics in architecture. Keywords: mathematics, geometry, science, number theory, architectural theory, digital architecture. iv ÖZ SAYILARDAN BASAMAKLARA: MATEMAT İĞİ N M İMAR İDE DE ĞİŞ EN ROLÜ ÜZER İNE Koç, Betül Yüksek Lisans., Mimarlık Bölümü Tez Yöneticisi: Doç. Dr. Zeynep Mennan Haziran 2008, 156 sayfa Bu çalı şma mimarlı ğın, matematik ve geometri ile pratik bir araç ve teorik bir referans olarak kurdu ğu ili şki üzerine ele ştirel bir incelemedir. Matematik ve matemati ğin metodolojik yapısının mimarlık söyleminin ilk olu şum dönemlerinden itibaren nihai bir temel ve güçlü bir referans oldu ğu iddia edilmektedir. Fakat, bu nihai temel ve bu temelin epistemolojik çıkarımlarının dayana ğı olarak kabul edillen matematiksel açıklık, metodolojik kesinlik, ve araçsal duyarlılık, sayısal(digital) teknolojilerin mimarlık prati ği içerisine girmesiyle yeni bir boyut kazanmı ştır. Bu teknolojilerin kullanımının fiziksel gerçekli ğin ya da uzamın geometrik temsili ve matematiksel kontrolüne dair önermi ş oldu ğu yeni olu şumlar, mimarlı ğın matematik disiplini ile olan ili şkisi içerisindeki dönü şüm noktalarının kuramsal çercevesi özelinde tartı şılacaktır. Anahtar Kelimeler: matematik, geometri, bilim, sayı teorisi, mimarlık teorisi, sayısal mimarlık. v ACKNOWLEDGEMENTS I would like to express my deepest gratitude to Assoc. Prof. Dr. Zeynep Mennan, for her patience, professional guidance, suggestions and encouragement, which guided me not only in the span of the thesis but also formed the most essential advises of my ongoing studies. I am also indebted to my family for their encouragement and support; and I owe a particular dept to my father for his thought stimulating critics and suggestions. The discussions that we have made and the quarrels that we have had either on philosophy or on architecture are the most essential references of my life and in specific this particular research. vi TABLE OF CONTENTS ABSTRACT ......................................................................................................iv ÖZ ....................................................................................................................... v ACKNOWLEDGEMENTS ..............................................................................vi TABLE OF CONTENTS .................................................................................vii LIST OF FIGURES .......................................................................................... ix CHAPTERS 1) INTRODUCTION .........................................................................................1 2) FROM THE TRANSCENDENTAL TO THE INSTRUMENTAL USE OF NUMBER ......................................................................................................... 13 2.1) Abstract Deductive Reasoning and the Birth of the Mathematical Spirit …………………………………………………………………………17 2.1.1) From a Practical Tool to a System of Thought ............................ 17 2.1.2) Reasoning and the Strength of Mathematical Construction.......... 19 2.1.3) An Acquired Universality ...........................................................20 2.2) Mathematics and the Cosmic Order ....................................................22 2.2.1) Mathematics as the Order and the Essence of Natural Phenomena ……………………………………………………………………22 2.2.2) The Value of Abstract Ideas and Sense Experience ..................... 32 2.3) Mathematical Grounds of Architecture until the 19 th Century ............. 35 2.3.1) Greek Architecture: Theory of Proportion as the Basic Aesthetic Element ……………………………………………………………………35 2.3.2) Renaissance: the Objective Existence of Mind and the Mathematization of Space ..........................................................................43 2.3.3) 17 th Century Architecture: The Rise of Empiricism and Observation................................................................................................ 61 2.4) A Turning Point in Mathematics.........................................................75 vii 2.4.1) The 17 th Century Mathematics: From Sensual to Intellectual Faculties ……………………………………………………………………75 2.4.2) 19 th Century Developments in Mathematics: a Crisis on the Foundations of Mathematics.......................................................................77 3) TOWARDS AN INSTRUMENTAL USE OF MATHEMATICS .............. 82 3.1) Interdisciplinarity ...............................................................................82 3.1.1) A new Platform of Exchange and Cross-fertilization for Architecture ............................................................................................... 82 3.2) Privileged Role of Abstract Information over Material Reality ........... 90 3.2.1) Uniformity between Architect’s Ideation/Intention and Numerically Controlled Architectural Production.......................................90 3.2.2) Surmounting the Limits of Human Reason and Intention ............ 95 3.2.3) Mathematical Relations as the Necessary Provision of a Digitally Uniformed Process.....................................................................................99 3.3) The Architectural Production as Intellectual Activity........................ 103 3.3.1) Self Organizing Systems: Extending the Limits of Formal or Functional Approaches in the Design Process........................................... 103 3.3.2) Generative Systems: From Determinate Structures to Indeterminate Systems …………………………………………………………………..111 3.4) The Architectural Process as Information Management .................... 125 3.4.1) Diagrams: Extending the Limits of Vision and Visual Representation ......................................................................................... 125 3.4.2) Diagrams: Merging Content and Expression ............................. 131 4) CONCLUSION .......................................................................................... 140 BIBLIOGRAPHY .......................................................................................... 146 viii LIST OF FIGURES Figure 1.1: Depiction of the superlunaryworld consisted of several circles and spheres. Source: Alexander Roob. The Hermetic Museum: Alchemy & Mysticism. Köln: Taschen, 2006, pp. 49-51. Originally in; A. Cellarius. Harmonia Macrocosmica, Amsterdam, 1660......................................................................23 Figure 1.2: Engravings that depict Pythagorean studies on the mathematical and geometric harmony of the universe and the first attempts on the way to construct mathematical concepts as independent, abstract entities. Source: Robert Lawlor. Sacred Geometry: Philosophy and Practice . London: Thames & Hudson, 2002, p. 7. ......................................................................................................................