Polyhedra and Beyond: Contributions from Geometrias’19, Porto, Portugal, September 05-07 (Trends in Mathematics)

This volume collects papers based on talks given at the conference "Geometrias'19: Polyhedra and Beyond", held in the Faculty of Sciences of the University of Porto between September 5-7, 2019 in Portugal. These papers explore the conference's theme from an interdisciplinary standpoint, all the while emphasizing the relevance of polyhedral geometry in contemporary academic research and professional practice. They also investigate how this topic connects to mathematics, art, architecture, computer science, and the science of representation. Polyhedra and Beyond will help inspire scholars, researchers, professionals, and students of any of these disciplines to develop a more thorough understanding of polyhedra. Foreword Preface Acknowledgments Contents List of Figures List of Tables Contributors About the Editors Chapter 1: Synthetic Methods for Constructing Polyhedra Introduction The Synthetic Approach for the Constrction of PS and AS The General Synthetic Method to Construct a Polyhedron Conclusions References Chapter 2: Scientific Sources and Representations of the Small Stellated Dodecahedra Painted in Genoa The Frescoes of Palazzo Balbi Senarega The Small Stellated Dodecahedron: Scientific and Iconographic History The Small Stellated Dodecahedron in the Room of Leda: Geometric Characteristics and Symbolic Meaning Observations on Shadows Conclusions References Chapter 3: Polyhedral Transformation Based on Confocal Quadratic Surface Properties. Graphical Speculations Introduction Generalization of One Property of the Archimedes Paraboloid Area of Application, Objectives, and Methodology From Plane to Space. Graphic Methods The Method of the Flat Polygonal Patterns The Method of the Circumference Mesh Discussion. Beyond the Stereographic Projection From Spheres to Other Quadrics. Projecting by Cones Projecting by Ellipsoids, Paraboloids, or Hyperboloids Discussion. Graphical Characterization of 3D Homology Conclusions References Chapter 4: Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort Introduction Investigating a Possibility for the Deltahedral Rings’ Formation Findings of the Research Conclusions References Chapter 5: Filling Space with Gyroid Symmetry Minimal Surfaces Discrete Minimal Surface The Gyroid Surface A Discrete Gyroid Surface Space-Filling Solids Filling Half-Space A. H. Schoen’s M6 Surface Cutting at Smooth Gyroid Surface Conclusions References Chapter 6: Odd or Even, Jitterbug Versus Grünbaum’s Double Polyhedra Introduction Jitterbug Transformations Applied to Non-Convex Polyhedra Face-Doubling Jitterbug Transformation Applied to Infinite Uniform Polyhedra Conclusion References Chapter 7: From Geometry to Reality: Designing Geodesic Structures Introduction The Subdivision Methods Study Case: The GoodKarma Constructive Method Comparing the Results Conclusions References Chapter 8: Vittorio Giorgini’s Architectural Experimentations at the Dawn of Parametric Modelling Introduction Giorgini as a Morphologist-Spatiologist Architect Giorgini as a Pioneer of Parametric Design Giorgini Parameterized Conclusions References Chapter 9: Architectural Inversions: The Intangible Aspect as a Form-Finding Factor in the Combined Work of Antoni Gaudí and John Pickering Introduction Design Process in Gaudí’s Later Years: 1914–1926 Ruled Surfaces in Design: Hypar & Hyperboloid of Revolution of One Sheet Complexity in the Sagrada Familia: Computational Thinking & Boolean Logic Mathematical Inversion as Form-Finding Strategy: John Pickering A Combined Approach to Virtual Presence Fabrication Strategies: Advantages and Constraints Fabrication Strategy 1: Unrolling Surfaces to Small Panels Using Dual Graphs (186 Panels) Fabrication Strategy 2: Unrolling Surfaces into Long Panels Defined by Geodesic Lines (60 Panels) Historical Traces of Gaudí’s Method to French Baroque Construction Conclusions References Chapter 10: An Introduction to Solid Tessellations with Students of Architecture Introduction Polyhedra and Solid Tessellations The Assignment Phase 1: Modelling the Tessellation Phase 2: Incorporating the Polyhedral Composition within the garden’s Context Phase 3: Definition of the Structure’s Materiality Phase 4: Presentation of the Project A Selection of the Projects Developed by the Students Project a: Stopping Point Project B: Through the Gradient Project C: (In)Tangible Conclusions References