Handbook of the mathematics of the arts and sciences. Volume 1

The goal of this Handbook is to become an authoritative source with chapters that show the origins, unification, and points of similarity between different disciplines and mathematics. Some chapters will also show bifurcations and the development of disciplines which grow to take on a life of their own. Science and Art are used as umbrella terms to encompass the physical, natural and geological sciences, as well as the visual and performing arts. As arts imagine possibilities, science attempts to generate models to test possibilities, mathematics serves as the tool. This handbook is an indispensable collection to understand todays effort to build bridges between disciplines. It answers questions such as: What are the origins of interdisciplinarity in mathematics? What are cross-cultural components of interdisciplinarity linked to mathematics? What are contemporary interdisciplinary trends? Section Editors: Michael J. Ostwald, University of Newcastle (Australia) Kyeong-Hwa Lee, Seoul National University (South Korea) Torsten Lindström, Linnaeus University (Sweden) Gizem Karaali, Pomona College (USA) Ken Valente, Colgate University, (USA) Consulting Editors: Alexandre Borovik, Manchester University (UK) Daina Taimina, Independent Scholar, Cornell University (USA) Nathalie Sinclair, Simon Fraser University (Canada) What do figure skating, invasive species, medieval cathedrals, ropes, poems, wines, metaphors, rhythms, climate change, and origami have in common? Mathematics! The Handbook of the Mathematics of the Arts and Sciences is a stunning compendium of essays on these and scores of other unlikely subjects to which the mathematical imagination has been brought. It is at once a dazzlingly contemporary tour of human success at bringing order to the world, and a throwback to a time before the "unity of knowledge" became a mere slogan. It is a breathtaking work, for its ambitious scope and for its endless stimulation of the reader's curiosity. - Harry R. Lewis, Gordon McKay Research Professor of Computer Science at Harvard University, editor of Ideas That Created the Future: Classic Papers of Computer Science Mathematics has always enjoyed deep connections with the arts, science, the humanities, philosophy, history, and society in general. However, these links are often overlooked or undervalued. This Handbook makes a massive statement about the extent and importance of the interdisciplinary nature of mathematics, and its relevance to all aspects of human culture. Its articles are scholarly and authoritative, but also highly readable and accessible to non-specialists. A triumph! -Ian Stewart FRS, Emeritus Professor - University of Warwick This handbook will delight anyone who loves the richness of mathematics and its interplay with the arts and humanities. Bharath Sriraman has given us a great gift, a treasure chest of connections to art and architecture, language and literature, philosophy, history, society, you name it. The book is authoritative and charming and instantly establishes itself as a landmark reference for interdisciplinary mathematics. Steven Strogatz - Jacob Gould Schurman Professor of Applied Mathematics at Cornell University.-- Provided by publisher Foreword Contents About the Editor Editorial Board Section Editors Consulting Editors Contributors Part I Mathematics, Art, and Aesthetics 1 Mathematics, Art, and Aesthetics: An Introduction References 2 The Art of Modern Homo Habilis Mathematicus, or: What Would Jon Borwein Do? Contents Introduction: Who Is Modern Homo Habilis Mathematicus? What Would Jon Do? Phase Portraits: A Motivating Example Reinvention by Bridging Between Contexts Repurposing Phase Portraits for Dynamical Systems Dynamical Geometry and Asymptotic Destination Plotting Experimentally Checking Numerical Error Completing the Circle: The Line from Specific to General Sometimes All You Need Is a Good Walk Walking on a Dynamical System When the Computer Knows More Than You Do Symbolic Answers from Numerical Approximations Conic Programming and Mystery Geometry Conclusion References 3 The Beauty of Blaschke Products Contents Introduction Complex Arithmetic and Geometry Seeing Complex Functions Hyperbolic Geometry Blaschke Products Blaschke Products and Ellipses in the Euclidean Plane Blaschke Products and Ellipses in the Poincaré Disk Model Compositions of Blaschke Products Conclusion References 4 Looking Through the Glass Contents Introduction A Brief History A New Mathematical Object: The Point of Projective Geometry Ideal Points Vanishing Points Where Was the Camera? A Consequence of Viewing Distances: Illusion, Distortion, and Anamorphism Dolly Zoom Anamorphic Art Impossible Figures Going Backward from Pictures to 3D Homogeneous Coordinates Multiple View Geometry The Ames Room Reconstructing Objects from Images Conclusion Cross-References References 5 Designing Binary Trees Contents Introduction Creating Binary Trees Mathematical Approach A First Example: L~LR A Second Example: LR~RL A Third Example: LR∞ ~ (RL)∞ Other Issues Artistic Considerations Conclusion References 6 Homeomorphisms Between the Circular Disc and the Square Contents Introduction Canonical Mapping Space Mapping Diagram with Equations Some Mathematical Details Fernandez-Guasti Squircle Tapered2 Squircular Mapping Lamé Squircle Elliptical Grid Mapping Conformal Square Mapping via Schwarz-Christoffel Legendre Elliptic Integrals A Fundamental Conformal Map Canonical Alignment Software Implementation A Complex Class of Squircles Application: Squaring the Poincaré Disk Hyperbolic Tilings Application: Elliptification of Rectangular Imagery Size Versus Shape Distortions Conclusion Cross-References References 7 A Visual Overview of Coprime Numbers Contents Introduction Coprime Numbers and Skew Sturmian Sequences Bézout Coefficients Ford Circles and Farey Sequences Bézout Graphs Conclusion References 8 Almost All Surfaces Are Made Out of Hexagons Contents Introduction Closed Surfaces Pants Decomposition Hyperbolic Plane and Negative Curvature Each Surface Admits More Than One Geometric Shape References 9 Anamorphosis Reformed: From Optical Illusions to Immersive Perspectives Contents Introduction Anamorphosis Formed Again The Empirical Principle: Radial Occlusion Anamorphosis Formed Fast Some Considerations on Anamorphosis The Point of Observation Multiple Points of Observation ``Impossible'' Objects On Color Binocular Anamorphoses Anamorphosis Formally Reformed Mathematical Preliminaries Anamorphosis as a Mathematical Object More General Surfaces Simplifications: Talking to Artists On Compactness Descriptive Geometry Construction of Anamorphoses Handmade vs Digital Anamorphoses Dürer Machines Running Back and Forth Perspectives Spherical Perspectives The Problem with Perspective Euclid and Psychophysics Leonardo's Axiom and Paradox Effects on the Development of Spherical Perspective Conclusion Cross-References References 10 Anamorphosis: Between Perspective and Catoptrics Contents Introduction Anamorphosis Between Paris and Rome: A Catoptric Relationship The Project for a Scientific Villa in Baroque Rome as a Mirror of Time Conclusion References 11 Geometric and Aesthetic Concepts Based on Pentagonal Structures Contents Introduction Tessellations and Their Dualizations Tiling with Regular Pentagons Pentagrid as Art Repertoire From the Pentagrid to the Kite-Dart-Grid Spatial Structures with Dodecahedra Spatial Structures with Rhombohedra: Golden Diamonds Geometry and Art: Reflections on Aesthetics Conclusion Cross-References References 12 Mathematics and Origami: The Art and Science of Folds Contents Introduction Modern Origami and Mathematical Axiomatization Origami and the Delian Problem Modular Origami Origami and Technology Art of Origami Conclusion Cross-References References 13 Geometric Strategies in Creating Origami Paper Lampshades: Folding Miura-ori, Yoshimura, and Waterbomb Tessellations Contents Introduction Background on Paper Lanterns Contemporary Origami-Inspired Paper Lampshades Light, Origami Design, and Material Design Parameters and Considerations for Origami Lampshade Design Flat-Foldable Origami Tessellations: Miura, Yoshumura, and Waterbomb Patterns Mathematical Theorems Governing Flat-Foldable Origami Tessellations Miura-ori Tessellation Miura-ori and the Bird's-Foot Vertex Folding Miura-ori into Cylindrical Lampshade with Translation Symmetry Folding Miura-ori into a Lampshade with Rotational Symmetry Yoshimura Tessellation Yoshimura Tessellation and Its Double Bird's Foot Vertex Folding Yoshimura into Cylindrical Lampshade with Translational Symmetry Folding Yoshimura into a Lampshade with Rotational Symmetry Waterbomb Tessellation Waterbomb Tessellation and Its Vertices Folding Waterbomb Tessellation into Cylindrical Lampshade with Translational Symmetry Folding Waterbomb Tessellation into a Lampshade with Rotational Symmetry Conclusion Cross-References References 14 Mathematical Design for Knotted Textiles Contents Introduction: Mathematics and Textiles Textile Knot Practice to Be Analyzed What is a Knot? Knot Theory and Its Diagrammatic Method Comparison Between Textile Knot Practice and Mathematical Knot Theory Analysis of Textile Knot Practice Using Knot Theory New Knot Pattern Designs Based on Knot Diagrams Use of New Materials Inspired by Knot Theory Analysis of Textile Knot Practice Using Braid Theory Definition of Tilings Analysis of Textile Knot Practice Using Tilings New Pattern and Structure Designs Based on Tiling Concepts Conclusion References 15 Art and Science of Rope Contents Introduction Terminology Archaeological and Historical Aspects Pottery Mosaic Materials Natural Fibers Plants Animalia Minerals Man-Made Fibers Methods of Construction Laid Rope Hand-Operated Equipment and Tools Machines Braided Rope Hand-Operated Equipment and Tools Machines Rope Properties Mathematical Properties Cross Section Rope Diameter Core Diameter Mechanical and Physical Properties Degree of Twist Linear Density Breaking Force Elongation Rope Length Conclusion References 16 A Survey of Cellular Automata in Fiber Arts Contents Introduction Cellular Automata Representations of Cellular Automata in Fiber Arts Sierpiński Triangles and Related Cellular Automata Other Designs from Well-Known Cellular Automata Rules Cellular Automata Designs Created for Fiber Arts Conclusion Cross-References References 17 Mathematics and Art: Connecting Mathematicians and Artists Contents Introduction Mathematical Tools for Artists Symmetry Asymmetry Mathematical Artists and Artist Mathematicians Geometrical Art Polyhedra, Tilings, and Dissections Origami Bridging the World of Art and Mathematics End Notes References 18 Mathematics and Art: Unifying Perspectives Contents Introduction Mathematics in Art Mathematics as an Artistic Inspiration Mathematics as an Artistic Tool and Medium The Interplay of Art, Culture, and Mathematics Artistic Ideas in Mathematics Graphs and Their Visualizations Examples of Graphs Knots and Graphs Reconfiguration Systems Unifying Perspectives Conclusion Cross-References References 19 Spherical Perspective Contents Introduction History Spherical Anamorphosis Radial Occlusion and Mimesis Spherical Anamorphs and Their Vanishing Points Spherical Perspective as Cartography of the Visual Sphere Referentials Azimuthal Coordinate System Horizontal Coordinate System Angular Measurements Azimuthal Equidistant Spherical Perspective (360-degree Fisheye) The Azimuthal Equidistant Flattening Solving the Azimuthal Equidistant Spherical Perspective Fixed Grids for the Azimuthal Equidistant Perspective A Ruler and Compass Construction of the Azimuthal Equidistant SphericalPerspective Perspective Constructions Tiled Floor (Central) Inside a Cube Arbitrary Square Tiled Floor Dynamic Grids Drawing from Nature Equirectangular Perspective VR Panoramas as Immersive Anamorphoses Construction of the Equirectangular Flattening Images of Geodesics Ruler and Compass Approximations Drawing Lines Sliding Grids Spherical Straightedges in Digital Drawing Programs Conclusion: What Is (Not) a Spherical Perspective Cross-References References 20 A Hidden Order: Revealing the Bonds Between Music and Geometric Art – Part One Contents Introduction Harmony Harmony of Time Harmony of Space The Pentagon The Hexagon The Octagon A Mapping between Music and Geometric Art Color to Pitch Relationship Loudness and Brightness Hue and Pitch Brightness, Loudness, and Pitch Timbre and Saturation A Relationship Between Rhythm and Pattern A Unit of Time and a Unit of Space Binary Counting Grid An Alternative Square Tiling Hilbert Curve Tiling The Dragon Curve Hexagons More Hexagons Rhythmic Motifs Rotations Grid Symmetry, Time Signature, and Structure of the Composition Pentagonal Symmetry Fibonacci, Bar Length, and Structure of Composition Indexing the Penrose Tiling Octagonal Symmetry Summary References 21 A Hidden Order: Revealing the Bonds Between Music and Geometric Art – Part Two Contents Structure of Final Design Indefinite Growth Linear Layout Change of Time Signature Compound Grids Performance Dynamics and Accents Timbre and Texture Creative Implications of a Translation Between Music and Art Creative Approaches Explored in A Hidden Order Applying Musical Composition Techniques to Geometric Artwork Introduction/Contrasting Sections Aperiodic Rhythms Conclusion Some Final Thoughts on the Research Pain and Gain Through Restrictions A Multidimensional Artistic Object Time as Space Looking Ahead Cross-References References 22 Korean Traditional Patterns: Frieze and Wallpaper Contents Introduction Frieze Patterns Wallpaper Patterns Some Designs Conclusion References 23 Projections of Knots and Links Contents Introduction Terminology Mathematical Concepts Geometry Knot Theory Knotwork Concepts Rectangular Diagonal Knotwork Circular Knotworks Turk's Head Archaeological and Historical Aspects Contemporary and Traditional Art Knotwork Analysis The Number of Components The Number of Crossings Braiding Pattern Symmetry Coloring Construction of Knotworks Discussion References 24 Comparative Temple Geometries Contents Introduction Islamic Region and Religion Trading Mathematics and Art Islamic Mathematics Islamic Geometric Patterns and Art Japanese Mathematics Japanese Temple Geometry Conclusion Cross-References References 25 Wasan Geometry Contents Introduction Wasan Wasan Geometry Problems Involving Congruent Circles Congruent Circles on a Line and a Circle Congruent Circles on a Line with Two Congruent Circles on a Line Congruent Circles on a Line and Congruent Squares Two Congruent Circles on a Line Congruent Circles on a Line with Two Intersecting Congruent Circles Two Sets of Congruent Circles on a Line and Two Circles A Square and Three Congruent Circles in an Isosceles Triangle Congruent Circles in a Rectangle The Arbelos in Wasan Geometry Two Sangaku Problems Involving a Circle of the Same Radius Two Congruent Circles Touching a Perpendicular to AB Two Circles Touching a Perpendicular to AB at the Same Point Two Congruent Circles Touching an Inclined Line to AB Congruent Circles Touching a Circle Passing Through the Center of α Reflection in the Axis Golden Arbelos Arbelos with Overhang Arbeloi Determined by a Chord A Sangaku Problem Involving an Archimedean Circle A Sangaku Problem Involving Two Archimedean Circles Wasan Geometry and Division by Zero The Configuration A(1) A Three-Circle Problem Practical Side Study of Wasan Geometry: Past and Present References 26 Geometries of Light and Shadows, from Piero della Francesca to James Turrell Contents Introduction Piero della Francesca's Darkness James Turrell's Darkness Conclusion References 27 TOND to TOND: Self-Similarity of Persian TOND Patterns, Through the Logic of the X-Tiles Contents Introduction: The Two Traditional Persian Families of Pentagonal Patterns The Kond + Sholl Family The Tond Family Multilevel Patterns. Reminders, and a New Case Two Kond Self-Similar Systems A Third Type of Kond Self-Similar System Transitions Between Different Families [A] == > [A] (from Kond + Sholl to Kond + Sholl) [A] == > [B]. From Kond + Sholl to Tond or, More Often, from [A1] to [B] [B] == > [A1]. From Tond to Kond Generalization of the First Example Generalization of the Second Example [B] == > [B]. From Tond to Tond X-Tiles Definition The X-Tiles and the Tond Traditional Family of Pentagonal Patterns Transition from Kond to Tond with the X-Tiles Tond to Tond Transition Through the X-Tiles Self-Similarity of TOND Patterns Through the X-Tiles Principle First Inflation Rule: System V1 The Inflation Rule Order of Appearance of the Tiles The Two-Level Tiles Second Inflation Rule: System V2 The Inflation Rule The Set of All the Tond Tiles that Can Emerge from the V3 System Order of Appearance of the Tiles The Two-Level Tiles Remark: Other Valid Orientation Options in the V2 System Third Inflation Rule: System V3 The Inflation Rule The Set of All the Tond Tiles that Can Emerge from the V3 System Option V3.1 The Two-Level Tiles and the Interlacings Option V3.4 Fourth Inflation Rule: System V4 The Inflation Rule Working with Decorated Rhombuses To Go Further Conclusion Cross-References References 28 Artistic Manifestations of Topics in String Theory Contents Introduction Glimpses into String Theory Genesis First Superstring Revolution Second Superstring Revolution AdS/CFT Correspondence The Imagery of String Theory A Piece of String Pants Diagram Calabi-Yau M.C. Escher Music Film and Television Ceramics Inspired by String Theory Circle Cusp Sewing Threehalves Cut Anomaly Subsurface Conclusions References 29 Cutting, Gluing, Squeezing, and Twisting: Visual Design of Real Algebraic Surfaces Contents From Algebraic Formulas to Geometric Forms: Real Algebraic Surfaces Standard Constructions: Union, Intersection, and Smoothing Morphing Symmetry Cutting and Gluing Squeezing, Shifting, and Twisting References 30 Double Layered Polyhedra Contents Elevation Vertex Figure Knots Holes and Compounds Connected Holes Connecting the Knots Odd or Even, Grünbaum's Double Polyhedra Versus Jitterbug Face-Doubling Jitterbug Transformation Applied to Infinite Uniform Polyhedra Unfolding Multilayer Polyhedra Unfolding the Double Layered Cube Double Layered Tetrahedron Double Layered Cuboctahedron Double Layered Dodecahedron Double Layered Icosahedron Elevation: Combinations of Polyhedra Strips and Rings Zonohedra Polar Zonohedra Conclusion Cross-References References Part II Mathematics, Humanities, and the Language Arts 31 Mathematics, Humanities, and the Language Arts: An Introduction Contents Cross-References 32 Mathematics and Poetry: Arts of the Heart Contents Introduction Mathematics of Poetry Syllabic Verse Rhyme Visual Form Other Mathematical Concerns About Poetry Poetry of Mathematics Poetic Mathematics Mathematical Poetry Educational Possibilities Further Reading and Making Connections References 33 ``Elegance in Design'': Mathematics and the Works of Ted Chiang Contents Introduction Direction Decryption Division Determination Writing Like a Heptapod: Nonlinear Semasiography Thinking Like a Heptapod: Variational Principles Premembering: Nonlinear Orthography and Nonlinear Time Story of Her Life Conclusion References 34 Running in Shackles: The Information-Theoretic Paradoxes of Poetry Contents Introduction The Form Paradox The Nonsense Paradox The Curious Case of Missing Synonyms A Word in Its Place Beyond Entropy Conclusion References 35 Metaphor: A Key Element of Beauty in Poetry and Mathematics Contents Introduction Beauty in Poetry and Math Metaphors in Mathematics A Taxonomy of Mathematical Metaphors Explicative or Homey Metaphors Discovery or Eureka Metaphors Creative or Special Metaphors Mathematical and Poetic Metaphors: Differences and Similarities Seven Differences Between Mathematical and Poetic Metaphors Seven Reasons Why Metaphor Creates Beauty (Emotion) in Poetry and Mathematics Cross-References References 36 Poems Structured by Mathematics Contents Introduction Early Examples of Mathematical Form The Oulipo and Raymond Queneau Sestinas Poetic Enumeration Syllables per Line Words per Line and Latin Squares Lines per Stanza and Pi Letters per Line Pantoums and Platonic Solids Fundamental Theorem of Arithmetic Poetry Incidence Geometry Poetics Summary and Concluding Remarks Cross-References References 37 Lewis Carroll's Defense of Euclid: Parallels or Contrariwise Contents Introduction Euclid and His Controversial Elements Emergence of Non-Euclidean Geometries Non-Euclidean Geometries and the Education System Charles Dodgson: The Oxford Mathematician Lewis Carroll's New Approach to the Euclidean Debate Geometric “Straight” Analogies Defense of the Parallel Postulate Carroll and Mathematics Examinations Euclid and His Modern Rivals Carroll's Misunderstandings of Non-Euclidean Geometries Conclusion: The Real Reason Carroll Fought for Euclid References Part III Mathematics and Architecture 38 Architecture and Mathematics: An Ancient Symbiosis Contents Introduction Relationships and Epistemology Mathematics in Architecture Mathematics for Architecture Mathematics of Architecture Conclusion Cross-References References 39 Egyptian Architecture and Mathematics Contents Introduction Definitions Accurate Reckoning for Enquiring into Things Scribes and Builders Mathematics and Architecture Practical Operations Meanings Beyond Numbers? Conclusions References 40 Labyrinth Contents Introduction Topology of Labyrinths Definitions Definition Mnemonic Devices Conclusion References 41 Classical Greek and Roman Architecture: Mathematical Theories and Concepts Contents Introduction The Figurate Representation of Quantities Arithmetic Geometry The Visual Comparison of Quantities The Theory of Proportion and Means Musical Proportions The Duplication of the Cube Art and Architecture Conclusion Cross-References References 42 Classical Greek and Roman Architecture: Examples and Typologies Contents Introduction Vitruvius Symmetry: Numbers and Ratios in Greek Temples Ionic Temples Doric Temples Arithmetization of Geometry Roman Innovation: Amphitheaters Conclusion Cross-References References 43 Mathematics and the Art and Science of Building Medieval Cathedrals Contents Abbreviations Introduction. The Cathedral and the Gothic Order Gothic Apses and Sacred Geometry The Theorica of the Canons of Tortosa Cathedral Commentary on Euclid's Elements by Al-Haijaj (c.325–c.265 BC) Saint Augustine's De Civitate Dei Translation of Plato's Timaeus by Calcidius, with Part of a Commentary Part of the Commentary on Plato's Timaeus by Calcidius Commentary on Somnium Scipionis by Macrobius Part of Geometria from Martianus Capella's Marriage of Philology and Mercury Geometria Incerti Auctoris by Gerbert (Silvester II) The Positional Number System of Adelard of Bath Practica Versus Theorica of Tortosa Cathedral The Construction of Heptagons The Construction of Octagons The Geometria Fabrorum Mathematics and the Art and Science of Building Medieval Cathedrals References 44 Renaissance Architecture Contents Introduction The Heritage from Classical Antiquity Mathematical Beauty in the Renaissance Beauty in Renaissance Architecture Perspective Conclusion Cross-References References 45 Baroque Architecture Contents Introduction Baroque Architecture and Architects Church Design: The Elongated Centrality Odd Polygons and Complex Curves Literary Sources and Onsite Studies Perspective and Anamorphosis Baroque Polymathy Conclusion Cross-References References 46 Temple of Solomon Contents Introduction Villalpando's Flawless System Ezechielem Explanationes' Influence Conclusion References 47 Utopian Cities Contents Introduction The Search for the Ideal City Conclusion References 48 Tessellated, Tiled, and Woven Surfaces in Architecture Contents Introduction Background to Tiling Tiling in Architecture Conclusion Cross-References References 49 Stereotomy: Architecture and Mathematics Contents Introduction Geometric Knowledge for the Rationalization of Structural Form Constructed with Small Elements Stereotomic Architecture Is Historically Based on Geometrical and Cutting Technique Knowledge The Application of Stereotomy Using Innovative Technology: “Stereotomy 2.0” Research About “Stereotomy 2.0” Stereotomy with 3D Printing in the Age of Industry 4.0 Conclusion Cross-References References 50 Fractal Geometry in Architecture Contents Introduction Background Fractal Geometry Fractal Geometry in Architecture Examples of Fractal Geometry in Architecture and Design Conclusion Cross-References References 51 Parametric Design: Theoretical Development and Algorithmic Foundation for Design Generation in Architecture Contents Introduction Generative Design Common Characteristics of Generative Design Main Generative Design Systems Generative Grammars Evolutionary Systems Emergent and Self-Organized Systems Associative Generation Parametric Design Historical Review of Parametric Design Origin of Parametric Design Development of Parametric Design Parametricism Parametric Design Reshaping Architectural Design Impact on Architectural Design Limitations of Parametric Design Conclusion Cross-References References 52 Shape Grammars: A Key Generative Design Algorithm Contents Introduction Background Basic Shape Grammars Main Components of a Shape Grammar Shape Grammar Application Designing a Shape Grammar Corpus Selection Shape Grammar Development Shape Grammar Evaluation Extensions of Basic Shape Grammars Parallel Grammars Parametric Grammars Graph Grammars Further Discussion on the Extensions Applications of Shape Grammars Description and Analysis Reproduction and Generation Optimization and Customization Combination with Other Methods Implementation of Shape Grammars Shape Grammar and Other Generative Design Algorithms Discussion and Conclusion References 53 Space Syntax: Mathematics and the Social Logic of Architecture Contents Introduction Space Syntax and Mathematics Spaces, Lines, and Points Application Conclusion Cross-References References 54 Isovists: Spatio-visual Mathematics in Architecture Contents Introduction Background Isovist Measures and Mathematics Application Conclusion Cross-References References 55 Fractal Dimensions in Architecture: Measuring the Characteristic Complexity of Buildings Contents Introduction Background The Box-Counting Method in Architecture Stage 1: Data Preparation Stage 2: Data Representation Stage 3: Data Preprocessing Stage 4: Data Processing Application Conclusion Cross-References References Part IV Mathematics in Society 56 Mathematics in Society: An Introduction 57 Probabilistic Thinking from Elementary Grades to Graduate School Contents Introduction Interpretations of Probability Probability in US Schools Probability in Grades K-12 Probability in Undergraduate Mathematics Measure-Theoretic Probability in Graduate Mathematics Subjective Probability in Graduate Mathematics Probabilistic Connections to the Sciences Conclusion Cross-References References 58 Risk and Decision Making: Modeling and Statistics in Medicine – Fundamental... Contents Introduction Rationality in Decisions in Health Issues Kinds of Thinking and Learning: Consequences of the Goal of Rationality Constituents of Risky Situations Nature and Definition of Risk Involved in Decisions Type of the Decision Situation People or Stakeholders Involved in the Decision The Quality of Information Risk Management in Health Issues The Difficulty to Assess Information Informed Consent Versus Shared Decisions Understanding Risk Statistical Methods in Medicine Significance Tests An Example Concerns with the P Value A Medical Diagnosis Based on Cut Points to Separate the Groups of Healthy and Ill An Analogy of the Medical Situation to Statistical Tests Sample Size Needed for Ensuring Good Quality of Information from Studies Conclusions Cross-References References 59 Risk and Decision Making: Modeling and Statistics in Medicine – Case Studies Contents Introduction Case Study 1: Risk Communication The Case of Lipitor: Absolute and Relative Risks Background Information The Advertising Campaign Is a Mixture of Objective Information and a Play with Emotions The Flaws of the Advertisement Campaign Absolute and Relative Risk and the Interpretation of Reducing Risks Empirical Evidence for the Claim of Superiority of Lipitor and the Risk Reduction Last But Not Least: The Missing Discussion About the Side Effects of Long-Term Medication Understanding the Statistical Information and Other Criteria for Judging the Risk Simplifying the Methods for Easier Communication and Understanding of Risks Case Study of Prostate Cancer Case Study of Breast Cancer Simplifying Supports the Communication But Introduces a Shift of Data Toward Facts Case Study 2: Dialogues on a Medical Diagnosis To Screen or Not to Screen A First Attempt to Compare Alternatives, Find Data, and Interpret the Risk Numbers First Investigations A Preliminary Evaluation of the Risk Further Data for a More Profound Evaluation of the Risk Prevalence: The Incidence of Breast Cancer is Dependent on Age An Interpretation of Correct-Negative: The Correct-Negative Rate Case Study 3: Benefits and Drawbacks of Screening Measuring the Success of Screening Programs Stakeholders Involved in the Introduction of Screening Programs Meta-Analyses: The Attempt of an Evaluation of Screening for Breast Cancer Increase in Lifetime and Number of Lives Saved Rate of False Positives Rate of False Negatives Evaluation of Potential Harm An Evaluation of the Impact of Screening as Compared to No Screening Success of Other Screening Programs Does the Evidence Support the Recommendations? Crucial questions for an informed decision are: Gigerenzer's Fact Box on Screening for Breast Cancer Gasche's Public-Health Discussion in Switzerland The US Discussion on Screening Conclusions Cross-References References 60 To Justice Through Statistics References 61 Actuarial (Mathematical) Modeling of Mortality and Survival Curves Contents Introduction to the Development and History of Mathematical Models of Mortality Life Insurance Before the Invention of the Mortality Table Importance of Having a Mortality Table The Innovation of Mortality Model De Moivre and the First Creation of a Mathematical Law of Mortality Gompertz and Makeham Laws of Mortality Other Parametric Mortality Models Stochastic Mortality Model for Individual Mortality Rate Joint Life Mortality Models Why Do We Need Joint Life Mortality Models? Copula Model A New Stochastic Mortality Model for Joint Lives Nonparametric Estimation of the Mortality Function One-Sample Estimation Joint Mortality Estimation Mortality Modeling with Cohort Effect Increasing in Human's Life Expectancy and Longevity Risk Lee-Carter Model Extensions of Lee-Carter Model Mitchell et al. (2013)’s Extension of the Mortality Model References 62 Mathematics in the Maritime Contents Introduction Calculating Latitude Calculating Longitude Map Making Global Positioning Systems The Least Squares Method The Advent of Insurance and Actuarial Science Conclusion Cross-References References 63 Mathematics and Economics, with Special Attention to Social Choice Theory Contents Introduction Mathematics in Economics, Game Theory, and Social Choice Theory General Equilibrium Theory Social Choice Theory Game Theory The Use of Mathematics in Economics Questioned Conclusion: The Indispensability of Mathematics References 64 Social Algorithms and Optimization Contents Introduction A Brief History Essence of Algorithms Optimization Algorithms Optimization Search for Optimality Advantages of Social Algorithms Social Algorithms Algorithms as Descriptive Systems Ant Colony Optimization Bees-Inspired Algorithms Algorithms as Linear Systems Particle Swarm Optimization Artificial Bee Colony Firefly Algorithm as a Nonlinear System Algorithms as Quasi-linear Systems Bat Algorithm Cuckoo Search Algorithm Analysis and Open Problems Algorithms and Self-Organization Balance of Exploitation and Exploration Open Problems Conclusions References 65 Applications of the Gini Index Beyond Economics and Statistics Contents Introduction Gini's Measures and the Lorenz Curve The Standard Deviation and Coefficient of Variation Applications of the Gini Index and GMD Society and Household Income Inequity Contrast in Grayscale Images Other Lorenz-Inspired Measures of Spread and Inequality Further Modeling with the Lorenz Curve and Gini Index Equalization and the Gini Index The Golden Equity Golden Academia Summary of Desirable Properties of Measures of Inequality and Spread Conclusions References 66 A Computational Music Theory of Everything: Dream or Project? Contents The World Formula: A Physical Theory of Everything (ToE) The ToE in Contemporary Physics Are Physicists Dreaming? Is ToE Essentially a Mathematical Problem? A Computational Music Theory of Everything (ComMute),a Mathematical Nightmare? Arguments Against a ComMute Individual Creativity Colonialist Universalism Uncontrollable Complexity What Does ``Computational'' Mean in ComMute? Some Directions Toward ComMute Two Dimensions, Same Idea: Harmony and Rhythm Understanding Harmony and Counterpoin