Handbook of the Mathematics of the Arts and Sciences

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 Counterpoint via Gestures
			Counterpoint Worlds for Different Musical Cultures
			Unification of Mental and Physical Realities in Music: Introducing Complex Time
			Unifying Note Performance and Gestural Performance: Lie Operators
			Unifying Composition and Improvisation?
		Conclusions
		References
	67 Groovy Mathematics: Toward a Theoretical Model of Rhythm
		Contents
		Introduction
			Order in Movement
			A Natural Attraction to Rhythmic Behavior and Experience of Rhythm
			Expressive Timing in Music
			Modeling Music Performance
		RFM: A Continuous Model of Rhythm Performance
			Oscillations and Rhythmic Structure
			Synthesis of Expressive Timing by Frequency Modulation
			Computer Implementation
		Simulating Movements in Rhythmic Behavior
			Synthesis of Asymmetric Movement Trajectories
			Illustration: RFM Simulation of fON
		Conclusion
		References
	68 Music, Dance, and Differential Equations
		Contents
		Introduction
		Music
			Sound Generation
			Musical Composition
		Dance
			Dance Movement
			Choreography
				Three-Body Problem
				Influenced by Chaos
				Choreography Using Waveforms
				Fluid Dynamics
				Movement of a Pendulum
		Summary
		Cross-References
		References
	69 Breaking the Ice: Figure Skating
		Contents
		Introduction
		History and Equipment
		Mathematics Within Skaters\' Blade Tracings
		Quantitative Ways to Describe Pattern Dances
		Geometric Transformations
			Rotations
			Reflections
			Translations
		Biomechanical Principles Within Skating
			Angular Momentum in Spins
				Moment of Inertia in Camel Spin
				Moment of Inertia in Upright Spin
				Conservation of Angular Momentum from Camel Spin to Upright Spin
				Is There Potential for More Record-Breaking Spins?
			Angular Momentum in Jumps
			Projectile Motion in Jumps
			Quintuple Jumps?
				Training Tools for Jumps
				Pole Harness
				Hinged Figure Skating Boot
				Weighted Gloves
		International Judging System Scoring
		Judging Biases
		Figure Skating Team Event
			Entrants\' Contributions to Their Team Scores
			Team Event Compared to Hypothetical Team Event
			Application of Hypothetical Team Event to Past Olympic Winter Games
		Summary
		References
	70 The Mathematical Foundations of the Science of Cities
		Contents
		Introduction
			Ebenezer Howard\'s Perspective on Cities
			Jane Jacobs\' Perspective on Cities
		Graph Theory
		Network Science
		Space Syntax
			The Axial Map
			Measures Using the Axial Map
			Criticisms of Space Syntax
		Road Network Analysis
			Named-Street Construction
			Intersection Continuity Negotiation
			Measures of Road Network Analysis
		Social Network Analysis
		Urban Scaling Theory
		Conclusions
		References
	71 Gilles Deleuze\'s The Fold: Calculus and Curvilinear Design
		Contents
		Introduction
		Deleuze\'s The Fold
		The Fold and Architecture
		Greg Lynn on Folded Architecture, Blobs, and Animate Form
		Summary
		Cross-References
		References
	72 Mathematics and Oenology: Exploring an Unlikely Pairing
		Contents
		Introduction
		Maths and Wine-Related Problems
			Barrel Volume Calculations
			The Mathematics of Wine Aging: Arrhenius and Eyring Equations
			Optimal Wine Storage Conditions
				Optimal Average Temperature
				Temperature Fluctuation
				Humidity
				Light
				Vibrations
			The Influence of the Heat Flow in the Temperature Equation
			The Optimal Depth for a Wine Cellar
			The Temperature Equation at the Optimal Depth
			A Qualitative Study of the Depth of a Wine Cellar Based on the Chosen Reference Period and Soil Conditions While the Temperature Is Changing
		What\'s Food and Wine Pairing?
			The Graph
			Geometrical Issues
			Matching Algorithm (MA)
			Implementation Details and Examples
		More Recent Investigations
		Conclusion
		References
	73 CombinArtorial Games
		Contents
		Introduction
			Rulesets
			Normal Play Games
			Computational Complexity
			Overview
			Heap Games
			Compounds of Games
			Aesthetics of Games
			Combinatorial Number Theory
			Play Games and Math Games
			The mex-Rule: a Minimal EXclusive Algorithm
		Three Games
			Fibonacci Nim
			Euclid\'s Game
			Wythoff Nim
			A Fibonacci Numeration System, ZOL
		Game Solutions
			Fibonacci Nim
			Euclid\'s Game
			Wythoff Nim
			Wythoff Properties
			Mex-Rule
			Floor-Function
			Fibonacci Morphism
			ZOL-Numeration
			Proofs of Solutions
			Proof for Fibonacci Nim
			Proof for Euclid\'s Game
			Proofs for Wythoff Nim
			Proof by Wythoff-Properties
			More on the Mex-Rule
			Proof of Floor-Function
			Proof of Fibonacci Morphism
			Proof of ZOL-Numeration
		When Sprague and Grundy Mex Bouton\'s Nim
			Sprague and Grundy Theory
		Conway\'s Theory of the Full Class of Normal Play
		Positional Games with Nonnegative Incentive
		Patterns of a Generalized Games
		Epilogue
		References
	74 Combinatorial Artists: Counting, Permutations, and Other Discrete Structures in Art
		Contents
		Introduction
		Combinatorics in Music
			Dodecaphonic Music
			Iannis Xenakis
			Tom Johnson
			Elliott Carter
			Further Examples
		Combinatorics in Literature
			The Oulipo
				Raymond Queneau
				George Perec
				Italo Calvino
			Juan Eduardo Cirlot
			Digital Poetry
				Brion Gysin
		Combinatorics in Visual Art
			Sol LeWitt
			Vera Molnar
			Manfred Mohr
			Vladimir Bonačić
			Anders Hoff Aka Inconvergent
			Other Combinatorial Visual Artists
		Dance, Theatre, and Cinema
			Dance
			Theatre
			Cinema
		Closing Time
		References
Part V Mathematics, Science, and Dynamical Systems
	75 Mathematics, Science, and Dynamical Systems: An Introduction
	76 Modern Ergodic Theory: From a Physics Hypothesis to a Mathematical Theory with Transformative Interdisciplinary Impact
		Contents
		Prelude
		Origins
		Consequence of the Ergodic Theorem and Other Significant Results
		Interdisciplinary Aspects of Ergodic Theory in Mathematics
			Number Theory
			Combinatorics
			Functional Analysis and Harmonic Analysis
			Fractal Geometry
		Interdisciplinary Aspects of Ergodic Theory with Other Disciplines
		References
	77 Two-Way Thermodynamics
		Contents
		Introduction
		Some Mathematics
		Opposite Arrows
		A Paradox
		Further Issues
		Conclusions
		Appendix: Precise Definition of the Modified ``Cat\'\'
		Notes
		References
	78 Visualizing Four Dimensions in Special and General Relativity
		Contents
		Introduction
		Mathematics of Space and Time
			Four-Dimensional Spacetime and the Special Theory of Relativity
			Gravity, Geometry, and the General Theory of Relativity
			Black Holes and Numerical Relativity
		Revealing Spacetime Through Technology
			Imagination and Artistry
			Analogies and Metaphors
			Spacetime Diagrams
			Relativistic Ray Tracing and First-Person Visualizations
			Gravitational Lensing and Astrophysical Observations
			Numerical Simulations of Gravitational Waves
			Virtual, Augmented, and Mixed Reality
		Conclusion
		Cross-References
		References
	79 Coevolution of Mathematics, Statistics, and Genetics
		Contents
		Introduction
		Early Contributions
			Mendel and His Inheritance Models
			Hardy-Weinberg Equilibrium
			Wright-Fisher Model
		Study of Family History and Pedigrees
			Twin Studies
			Genetic Linkage Mapping
		Exploring Big Genetic Data
			Genome-Wide Association Studies
			Whole Genome Sequencing
			Network-Based Analysis for Genetic Data
		Discussion
		References
	80 Topology in Biology
		Contents
		Introduction
		What and Why Topology?
		Finding Topological Cavities: Persistent Homology
		Data Systems and Solutions: Sheaves
		Lead-Lag Relationships: Path Signatures
		Where Are We Going?
		Citation Diversity Statement
		References
	81 Dynamical Systems and Fitness Maximization in Evolutionary Biology
		Contents
		Introduction
		Historical Development of Natural Selection and Genetics
			Charles Darwin and Survival of the Fittest
			Gregor Mendel and Experimental Genetics
			The Eclipse of Darwinism
			Population Genetics
			Fitness Maximization and the Neo-Darwinian Theory of Evolution
			The Decline of Fisher\'s Fundamental Theorem
		Fisher\'s Fundamental Theorem of Natural Selection
			Fisher\'s Setting for His Fundamental Theorem
			Fisher\'s Mathematical Model for His Fundamental Theorem
			Mutations and Fisher\'s First Corollary
			Genetic Variance and Fisher\'s Second Corollary
			Review of Fisher\'s Biological Setting for His Theorem
		The Problem of Genetic Mutation
			Muller and Muller\'s Ratchet
			Models of Selection and Mutation
			Mutation-Selection Models with More Realistic Factors
			Numerical Simulations from the FTNSWM Mutation: Selection Equations
			Conclusions from Mathematical Mutation-Selection Models
		Comprehensive Simulations and Comprehensive Fitness
			The Necessity of Comprehensive Numerical Simulations
			Other Challenges to Net Fitness Maximalization
			Why Have We Not Died 100 Times Over?
			Lewontin\'s Lamentations
			Reductive Evolution
		Evolutionary Models, Dynamical Systems, and Maximization Principles
			Stable Equilibria in Mutation-Limited, Infinite Population, Perfect Selection Scenarios
			Conley\'s Fundamental Theorem of Dynamical Systems
			Are There Laws in Biology?
		A Biological Experiment, Individual Mutations, Adaptation, and Fitness
			The Long-Term Evolutionary Experiment
			Mutation-Selection-Reproduction Experimental Results
			LTEE Experiment and Mathematical Modeling Conclusion
			Maximization of Net Biological Function
		Conclusion
			Skepticism of Fitness Maximization
		References
	82 Damped Dynamical Systems for Solving Equations and Optimization Problems
		Contents
		Introduction
		Linear Problems
			Linear Equations
			Linear Eigenvalue Problems
			Linear Least Squares
			Ill-Posed Problems
				Numerical Simulations
		From Linear to Nonlinear Problems
			Local Linearization Using Optimal Damping and Time Step
			Total Energy as a Lyapunov Function
			Numerical Experiments
		Applications
			Image Analysis
			Inverse Problems for Partial Differential Equations
				Numerical Simulations
			Applications in Quantum Physics
				Excited States to the Schrödinger Equation
				The Yrast Spectrum for Atoms Rotating in a Ring
				Phase Separation of Bosonic- and Fermionic-Densities in an Ultracold Atomic Mixture
		Conclusions and Future Work
		References
	83 Mathematics and Climate Change
		Contents
		Introduction
		Climate: A Fluid Dynamical System
			Mathematical Equations
			Nondimensional Parameters: The Reynolds Number
			Convection in the Rayleigh-Bénard System
			Reduction of Dimensions and the Lorenz System
			Scaling in the Climate System
			Projection Methods: Coarse Graining and Stable Manifold Theory
		Brownian Motion, Weather, and Climate
			Climate Variability and Sensitivity
			Non-normal Growth of the Climate System
			Predictability
		Boltzmann Dynamics
		Conclusions
		Cross-References
		References
	84 Mathematical Models Can Predict the Spread of an Invasive Species
		Contents
		Introduction
		Population Growth Models
		Dispersal by Diffusion
		Conclusion
		Cross-References
		References
	85 Mathematics and Recurrent Population Outbreaks
		Contents
		Introduction
		The Lotka–Volterra Model
		Advantages of the Lotka–Volterra Model
		Criticism Against the Lotka–Volterra Model
		Gause-Type Models for Population Interaction
		What About Real Chemostat Conditions?
		References
	86 Limit Cycles in Planar Systems of Ordinary Differential Equations
		Contents
		Introduction
		Planar Linear and Linearized Systems
		First Integral Systems and Gradient Systems
		Monotone Dynamics
		Index Theory
		The Complex Plane
		The Existence of Limit Cycles
		The 34:Lienard.Revue:23 Equation
		Theorems for Absence of Limit Cycles
		Uniqueness of Limit Cycles
		Summary
		References
	87 Mathematical Models in Neuroscience: Approaches to Experimental Design and Reliable Parameter Determination
		Contents
		Introduction
		Chemical Kinetics Schemes and the Law of Mass Action
		Characteristic Scales and Model Non-dimensionalization
		Brief Review of Asymptotic Analysis and Asymptotic Algorithm for Model Reduction
		Quasi-Steady-State Approximation and Michaelis–Menten–Henri Kinetics
		NMDAR Desensitization: Background Information and General Model
		Kinetic Model of NMDAR and Experiment Design
		Initial Conditions for NMDAR Experiments
		Reduction of the NMDAR Model in Case of Experiments with High Concentration of D-Serine
		Reduction of the NMDAR Model in Experiments with High Concentration of L-Glutamate
		Reduction of the NMDAR Model in Experiments with High Concentrations of D-Serine and L-Glutamate
		Reduction of the NMDAR Model After the Pulse
		Reliable NMDAR Model Parameter Estimation
		Model Fitting to Data
		Conclusion
		References
	88 Interdisciplinary Mathematics and Sciences in Schematic Ocean Current Maps in the Seas Around Korea
		Contents
		Introduction
		Direct Measurement and Indirect Estimation of Ocean Current
			In situ Measurement Using Instruments
			Surface Current from Satellite Altimeter Data
			Surface Current from Surface Drifters
			Maximum Cross Correlation Method from Sequential Satellite Images
		Navigation and Registration of Ocean Current Maps
			Unified Geographical Mapping Procedure
			Digitized Current Maps of Textbooks and Scientific Articles
		Strategy for Unified Current Map
			List of Topics and Issued Contents
			Working Flow for Finalized Schematic Map
		Schematic Map of Ocean Current
			Case I: East Sea (Japan Sea)
			Case II: Yellow Sea and East China Sea
		Other Issues
			Name of Current
			Use of Colors
			Use of Lines
			Strength of Current
			Quantitative Information on Digital Ocean Current Map
		Implications to Other Countries
		Conclusion
		References
Part VI Mathematics, History, and Philosophy
	89 Mathematics, History, and Philosophy: An Introduction
		References
	90 Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today
		Contents
		Introduction
		Traces of Mathematics of the First Humans
		History of Ancient Mathematics: The First Written Sources
		History of Mathematics or Heritage of Mathematics?
		Further Views of the Past and Its Relation to the Present
		Can History Be Recapitulated or Does Culture Matter?
		Concluding Remarks
		Cross-References
		References
	91 Mathematics and Cultures Across the Chessboard: The Wheat and Chessboard Problem
		Contents
		Introduction
		Mathematics and the Invention of Chess
		Mathematics and the Origins of Chess
		Geometric Progressions and Chess
		Arabic Sources on the Computation 264 − 1
		Greek Sources on the Computation 264 − 1
		Western Sources on the Computation 264 − 1
		Number Theory
		Summary
		Cross-References
		References
	92 Ancient Greek Methods of Measuring Astronomical Sizes
		Contents
		Introduction
		Conclusion
		Cross-References
		References
	93 Space and Time in the Foundations of Mathematics, or Some Challenges in the Interactions with Other Sciences
		Contents
		The Geometric Intelligibility of Space, an Introduction
			Euclid
			B. Riemann
			A. Connes
			Some Epistemological Remarks on the Geometry of Physical Space
		Codings
			Geometry in Computing
		Living in Space and Time
			Multiscale Phenomena and the Mathematical Complexity of the Neural System
		Theories Versus Models
		Conclusion: Epistemological and Mathematical Projects
			Epistemology
			Geometry in Information
			Geometric Forms and Meaning
		References
	94 Baroquian Folds: Leibniz on Folded Fabrics and the Disruption of Geometry
		Contents
		Introduction
		Folded Drapery: Between Geometry and Its Subversion
			Before the Baroque: The Geometrization of Folded Drapery
		Folds of the Baroque: Disruption of and Deviation from the Geometrical Space
		Leibniz on Folding
		Conclusion
		References
	95 Nyaya Methodology and Western Mathematical Logic: Origins and Implications
		Contents
		Introduction: Debate Over the Importance of Nyaya Philosophy
		Comparisons Between the Aristotelian Syllogism and Nyaya Syllogism
		Valid Knowledge and Logical Methods in the Nyaya System
		Flaws in the Law of Contrapositive
		Navya-Nyaya Theory of Number
		Aristotle v. Nyaya: Final Word
			The Nyaya Syllogism\'s Conceptual Origins and Implications
		Origins of Logic
		The Original Debate: Milinda-Panha
		Logical Objects
		Four-Cornered Negation
		Kathavatthu and the Vadayutti
		The Nyayasutra
		The Nyaya Syllogism and the Problem of Jati
		Summary/Conclusion
		References
	96 Reception and Contestation: Mathematics and Esoteric Spirituality, 1875–1915
		Contents
		Introduction
		Hyperspace Theorizing and Early Theosophical Interventions
		Contesting the Fourth Dimension
		Hyperspace in Ouspensky\'s Tertium Organum
		Making Sense of an Erratic Discourse
		Concluding Remarks
		References
	97 Islamic Design and Its Relation to Mathematics
		Contents
		The Geometric Mode in Islamic Art
		Theories, Problems, and Evidence
		Symbolic Meaning
		Early Islamic Art: The Emergence of an Islamic Aesthetic Sensibility
		Islam\'s Greek Inheritance: Mathematics, Science, and Philosophy
		Theoretical Geometry and Artisanal Practice in the Islamic World
		Mathematics in the Islamic World and Its Involvement in Geometric Ornament
		Conclusion of Historical Perspective
		Modern Mathematical Analysis
		Computer Usage
		References
	98 Mathematical Explanations and Mathematical Applications
		Contents
		Introduction
		Catastrophes and Games
		Curious Cicadas and Simple Strawberries
		What Are Mathematical Explanations Like?
		Philosophical Significance
		Conclusion
		References
Part VII Mathematical Influences and New Directions
	99 Introduction to Mathematical Influences and New Directions
		References
	100 Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices
		Contents
		Introduction
		Ethnomathematics and Modelling
		Exploring Ethnomodelling
		Ethnomodelling and its Three Approaches of Viewing Cultures
			Etic: The Global/Outsider Approach
			Emic: The Local/Insider Approach
			Dialogic: The Glocal/Emic-Etic Approach
		Characterizing Ethnomodels
			Emic and Etic Ethnomodels of the Mangbetu Ivory Sculpture
			An Etic Ethnomodel of Brazilian Roller Carts
			A Dialogic Ethnomodel of a Local Farmer-Vendor
		Relevance of Ethnomodelling in a Mathematics Curriculum
		Conclusion
		References
	101 Cognition, Interdisciplinarity, and Equity
		Contents
		Introduction
		Focus and Criteria for the Review
		Selected Works Influenced by Cultural Anthropology and Ethnography
			Ethnomathematics Research
			Funds of Knowledge Research
			Summary and Future Research: The Importance of Community Engagement
		Individual Cognition of Academic Mathematics
			Malloy and Jones (1998)
			Morton (2014)
			Adiredja (2019) and Adiredja and Zandieh (2017, in press)
			Lewis (2014) and Lewis and Lynn (2018a, b)
			Fuson, Smith, and Lo Cicero (1997)
			Summary and Future Research: Diversity in Engaging the Politics of Mathematical Learning
				The Use of Existing Literature
				The Recruitment of Participants and Emancipatory Approach
				Managing Generalization and Essentialization of Findings
		Conclusion
		Cross-References
		References
	102 Mathematics and Rhetoric
		Contents
		Introduction
		Why Study Math from a Rhetorical Perspective?
		What Do Mathematicians Have to Gain?
		Summary: How Is a Rhetorical Approach Different from Other Interdisciplinary Approaches?
		References
	103 Modes and Modalities of Mathematical Authority: Disseminating the ``New Infinite,\'\' 1870–1920
		Contents
		Introduction
		Mathematical Considerations
		In Advance of the New Infinite, 1870–1890
		Josiah Royce (1855–1916): The New Infinite and the Absolute
		Cassius J Keyser (1862–1947): Policing and Promoting the New Infinite
		Responses and Other Commentaries, 1900–1920
		Concluding Remarks
		Cross-References
		References
	104 ``Bok Bok\'\': Exploring the Game of Chicken in Film
		Contents
		Introduction
		Theoretical Chicken: Analyzing the Game
			Payoff Matrices and Non-zero-sum Games
			Rationality
			Cooperation Versus Defection
			Equilibrium Points
			Communication
		Applied Chicken: Winning Friends and Influencing People
			Why Play the Game?
			``Nobody Here but Us Chickens\'\'
			``Don\'t You Play Chicken with Me, Boy\'\'
			``Chickens Are Bitches, Dude\'\'
		Conclusion
		References
	105 Moral Mathematics
		Contents
		Introduction
		Dollar Auction Vignette
		History of Moral Math
		Limitations, Resistance, and Cautions
		Ten Examples of Math Used as a Tool to Impact Social Behavior
		Experiential Presentations
		Five (of Many) Areas to Target for Continued Moral Math Development
		Potential for Spiritual Healing
		Closing Vignette
		Conclusion
		References
	106 Feminist Theories Informing Mathematical Practice
		Contents
		Introduction
		Mathematics and the Shadow of Gender Essentialism
		Mathematics, Feminist Perspectives, and Connections to Science and Technology Studies
		Mathematics, Issues of Power, and Pedagogical Practice
		Mathematics, Popular Culture, and Representation
		Conclusion
		References
	107 Queer(y)ing Mathematical Knowledge and Practices
		Contents
		Introduction
		Appreciating Queer in Context
		Queering Visibility, Support, and Resources in Mathematics and STEM
		Queering Curricula
		Queer(y)ing Perspectives on Disciplinary Knowledge and Practices
			Alan Turing
			Reuben Hersh: What Is Mathematics, Really?
			Imre Lakatos: Proofs and Refutations
		Concluding Remarks
		References
Index