Thinking, Drawing, Modelling: Geometrias 2017, Coimbra, Portugal, June 16–18 (springer Proceedings In Mathematics & Statistics (326))

This book presents a selection of papers from the International Conference Geometrias’17, which was hosted by the Department of Architecture at the University of Coimbra from 16 to 18 June 2017. The Geometrias conferences, organized by Aproged (the Portuguese Geometry and Drawing Teachers’ Association), foster debate and exchange on practical and theoretical research in mathematics, architecture, the arts, engineering, and related fields. Geometrias’17, with the leitmotif “Thinking, Drawing, Modelling”, brought together a group of recognized experts to discuss the importance of geometric literacy and the science of representation for the development of scientific and technological research and professional practices. The 12 peer-reviewed papers gathered here show how geometry, drawing, stereotomy, and the science of representation are still at the core of every act leading to the conception and materialization of form, and highlight their continuing relevance for scholars and professionals in the fields of architecture, engineering, and applied mathematics. João Pedro Xavier is an Architect and Full Professor at the Faculty of Architecture,University of Porto (FAUP), where he received his initialdegree in 1985 and his Ph.D. in Architecture in 2005. From 1986 to 1999, he worked in Álvaro Siza’s office. At the same time, he established his own practice as an Architect. Xavier is a member of the CEAU research teams Architecture: Theory, Project, History (ATPH) and Digital Fabrication Laboratory (DFL). The relationship between architecture and mathematics, especially perspective, is his main research interest. He is the author of the books Perspectiva, perspectivaacelerada e contraperspectiva (Perspective, accelerated perspective and counter-perspective) and Sobre as origens da perspectivaem Portugal (On the origins of perspective in Portugal). He is a Correspondent Editor of the Nexus Network Journal and a Council Member atthe European Association for Architectural Education (EAAE). Vitor Murtinho is an Architect and Full Professor at the Department of Architecture,University of Coimbra (UC), Portugal. A Senior Researcher at UC’s Center for Social Studies, his research interests include Renaissance theory, architectonics of form, and geometry. As the UC’s Vice-Rector (March 2011 to February 2019), he was responsible for heritage, buildings, and sustainability. He is the author or co-author of more than one hundred publications (books, book chapters, and papers in journals and conference proceedings). A detailed list of his publications can be found at http://ces.uc.pt/en/ces/pessoas/investigadoras-es/vitor-murtinho/publicacoes. Vera Viana has been the Director of Aproged (the Portuguese Geometry and Drawing Teachers’ Association) since 2009 and organizes, among other events, the International Conferences “Geometrias.” She is Editor-in-Chief of the Geometrias Conference Proceedings and Aproged’s Bulletins, and serves on the scientific committees of several international conferences. As a Researcher at the University of Porto’sCentre for Studies of Architecture and Urbanism, her work focuses on the connections between architecture and mathematics. Since 2001, she has been engaged in the development of educational resources, particularly on polyhedral geometry with three-dimensional modelling and algorithmic modelling software, and has released a Ph.D. thesis, papers, and presentations on the subject. Preface 6 Acknowledgements 10 Contents 12 Prototyping Stereotomic Assemblies: Stone Polysphere 14 1 Background 14 2 Research Topic 15 3 Computational Workflow 15 4 Prototypation 19 5 Conclusions 20 References 23 Geometry and Digital Technologies in the Architecture of Herzog & de Meuron. The Project for the Stamford Bridge Stadium in London 25 1 Herzog & de Meuron 25 1.1 Project Practice 25 1.2 Project Theory 26 2 Stamford Bridge Stadium 27 2.1 Geometry 28 2.2 Digital Technologies 36 3 Conclusions 38 References 39 The Dome as Minimal Housing Unit: “Ghibli” and “D-Home” Prototypes 41 1 Introduction 41 2 Ghibli 42 3 D-Home 47 4 Conclusions 51 References 51 Geometry and Art 53 1 A Historical Debate 53 2 Geometry and Geometries 56 3 The Art of Geometry 57 4 The Theoretical Foundation of Geometry 59 5 Scienctia Scientiarum 61 6 Texts About Geometry 62 7 Practical Geometry 62 8 “I Will not Speak as a Mathematician but as a Painter” 63 9 The Geometric Order of Beauty 65 10 Geometry in the Royal Academies 69 11 The Current Landscape 70 References 71 The vaults of Arronches Nossa Senhora da Assunção and Misericórdia churches. Geometric and constructive comparison with the nave and refectory vaults of Jerónimos Monastery 73 1 Introduction 73 2 Cross-Ribbed Vaults—Geometry and Construction 75 3 Arronches Assunção Church—Lateral Chapel Vault 77 3.1 Design and Construction Hypothesis 77 4 The Nave Vault of Nossa Senhora Da Assunção Church 78 4.1 Comparison with the Vault of Jerónimos Monastery Church 78 4.2 Plan 79 4.3 Elevation 80 4.4 Construction 81 5 The Nave Vault of Arronches Misericórdia Church 81 5.1 Plan 81 5.2 Elevation 82 5.3 Construction 83 6 Jerónimos Refectory Vault 83 6.1 Plan 84 6.2 Elevation 84 6.3 Construction 85 7 Conclusions 85 References 86 Perspective Transformations for Architectural Design 88 1 From Perspective to Scenography 88 2 Mathematical Concepts of Vanishing Points and Transformation 91 3 From Scenography to Relief Perspective 92 4 Perception and Architectural Design 94 5 Examples and Experiments of Perspective Transformational Approaches in Architectural Design 96 6 Conclusions 98 References 98 Ordered Creativity: The Sense of Proportion in João Álvaro Rocha’S Architecture 101 1 Introduction 102 2 Methodology 102 3 Case Study #1—Paçô House, Viana Do Castelo (1994–1997) 104 4 Case Study #2—Tomé Sousa House, Porto (2001–2009) 110 5 Final Remarks 113 6 Conclusions 114 References 114 Developable Ruled Surfaces from a Cylindrical Helix and Their Applications as Architectural Surfaces 116 1 Developable Helical Surfaces 116 2 Evolute, Involute and Evolvent 117 3 Factors of the Development 118 3.1 Development by Circular Sectors 118 3.2 Developments of Helicoids Generated by Evolvents 119 4 Curvature of a Curve and Curvature of a Surface 119 5 Helicoids that Share the Same Development 121 5.1 Helicoids that Share the Same Planar Development 121 5.2 Family of Helicoids that Share the Same Development (Circular Sector) and Whose Generating Lines Are Tangent to the Helix 122 5.3 Development by Evolvents 122 6 Transformable Helicoidal Folded Structures 123 6.1 Folding on Concentric Rings 123 6.2 Folding Evolvents 126 7 Conclusions and Discussion 127 References 128 Porous Spatial Concrete Structures Generated Using Frozen Sand Formwork 130 1 Concept and Approach 131 2 Method 131 3 Prototype 133 4 Conclusion 134 References 137 Calculated Geometries. Experiments in Architectural Education and Research 139 1 Introduction 139 1.1 On Geometry in Architecture 139 1.2 The Descriptive Versus the Generative Approach 140 2 Calculated Geometries 140 2.1 The Analogical Condition 140 2.2 The Digital Condition 141 3 Experiments in Architectural Education and Research 141 3.1 Education: “Parametric Architecture”—IAAC, 2008 142 3.2 Research: “Trefoil Structure”—DFL, 2013 142 3.3 Education: “Parametric Constructions”—FAUP, 2010–16 144 3.4 Education + Research: “Robotic Assemblies”—FAUP, 2015–16 146 3.5 Education: “The Curved Building”—FAUP, 2010–16/DArq-FCTUC, 2010–12 147 4 Conclusion 149 References 150 How to Construct the Red Sea? 152 1 Introduction 152 2 Concept of the Museum 154 3 Concept of the Curvilinear Wall 155 4 Conclusions 157 References 159 How to Improve the Education of Engineers—Visualization of String Construction Bridges 161 1 Introduction 161 2 Creation 162 3 Education 162 4 Representation 164 5 Inspiration 165 6 Conclusion 166 References 166 Preface......Page 6 Acknowledgements......Page 10 Contents......Page 12 1 Background......Page 14 3 Computational Workflow......Page 15 4 Prototypation......Page 19 5 Conclusions......Page 20 References......Page 23 1.1 Project Practice......Page 25 1.2 Project Theory......Page 26 2 Stamford Bridge Stadium......Page 27 2.1 Geometry......Page 28 2.2 Digital Technologies......Page 36 3 Conclusions......Page 38 References......Page 39 1 Introduction......Page 41 2 Ghibli......Page 42 3 D-Home......Page 47 References......Page 51 1 A Historical Debate......Page 53 2 Geometry and Geometries......Page 56 3 The Art of Geometry......Page 57 4 The Theoretical Foundation of Geometry......Page 59 5 Scienctia Scientiarum......Page 61 7 Practical Geometry......Page 62 8 “I Will not Speak as a Mathematician but as a Painter”......Page 63 9 The Geometric Order of Beauty......Page 65 10 Geometry in the Royal Academies......Page 69 11 The Current Landscape......Page 70 References......Page 71 1 Introduction......Page 73 2 Cross-Ribbed Vaults—Geometry and Construction......Page 75 3.1 Design and Construction Hypothesis......Page 77 4.1 Comparison with the Vault of Jerónimos Monastery Church......Page 78 4.2 Plan......Page 79 4.3 Elevation......Page 80 5.1 Plan......Page 81 5.2 Elevation......Page 82 6 Jerónimos Refectory Vault......Page 83 6.2 Elevation......Page 84 7 Conclusions......Page 85 References......Page 86 1 From Perspective to Scenography......Page 88 2 Mathematical Concepts of Vanishing Points and Transformation......Page 91 3 From Scenography to Relief Perspective......Page 92 4 Perception and Architectural Design......Page 94 5 Examples and Experiments of Perspective Transformational Approaches in Architectural Design......Page 96 References......Page 98 Ordered Creativity: The Sense of Proportion in João Álvaro Rocha’S Architecture......Page 101 2 Methodology......Page 102 3 Case Study #1—Paçô House, Viana Do Castelo (1994–1997)......Page 104 4 Case Study #2—Tomé Sousa House, Porto (2001–2009)......Page 110 5 Final Remarks......Page 113 References......Page 114 1 Developable Helical Surfaces......Page 116 2 Evolute, Involute and Evolvent......Page 117 3.1 Development by Circular Sectors......Page 118 4 Curvature of a Curve and Curvature of a Surface......Page 119 5.1 Helicoids that Share the Same Planar Development......Page 121 5.3 Development by Evolvents......Page 122 6.1 Folding on Concentric Rings......Page 123 6.2 Folding Evolvents......Page 126 7 Conclusions and Discussion......Page 127 References......Page 128 Porous Spatial Concrete Structures Generated Using Frozen Sand Formwork......Page 130 2 Method......Page 131 3 Prototype......Page 133 4 Conclusion......Page 134 References......Page 137 1.1 On Geometry in Architecture......Page 139 2.1 The Analogical Condition......Page 140 3 Experiments in Architectural Education and Research......Page 141 3.2 Research: “Trefoil Structure”—DFL, 2013......Page 142 3.3 Education: “Parametric Constructions”—FAUP, 2010–16......Page 144 3.4 Education + Research: “Robotic Assemblies”—FAUP, 2015–16......Page 146 3.5 Education: “The Curved Building”—FAUP, 2010–16/DArq-FCTUC, 2010–12......Page 147 4 Conclusion......Page 149 References......Page 150 1 Introduction......Page 152 2 Concept of the Museum......Page 154 3 Concept of the Curvilinear Wall......Page 155 4 Conclusions......Page 157 References......Page 159 1 Introduction......Page 161 3 Education......Page 162 4 Representation......Page 164 5 Inspiration......Page 165 References......Page 166 M. Barberio, Prototyping Stereotomic Assemblies: Stone Polysphere -- L. Cabezas, Geometry and Art -- A. Castro, Geometry and Digital Technologies in Herzog & De Meuron Architecture -- M. Colella, The Dome as Minimal Housing Unit: Ghibli and D-Home Prototypes -- S. Genin, The Vaults of Arronches Nossa Senhora da Assunção and Misericordia Churches. Geometric and Constructive Comparison with the Nave and Refectory Vaults of Jerónimos Monastery -- C. Leopold, Perspective Transformations for Architectural Design -- J. Maia and V. Murtinho, Ordered Creativity: The Sense of Proportion in the Architecture of João Álvaro Rocha -- A. Martín-Pastor and A. Lopez Martínez, Developable Ruled Surfaces from a Cylindrical Helix and their Application as an Architectural Surface -- H. Müller, C. Nething, A. Schalk, D. Kovaleva, O. Gericke and W. Sobek, Porous Spatial Concrete Structures Generated Using Frozen Sand Formwork -- J. Pedro Sousa, Calculated Geometries. From Manual to Robotic Experiments in Architectural -- M. Sroka-Bizoń, How to Construct Red Sea? -- J. Tofil, How to Improve the Education of Engineers Visualization of String Construction Bridges.