Polyhedra and Beyond: Contributions from Geometrias’19, Porto, Portugal, September 05-07

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