Electronic String Art: Rhythmic Mathematics

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Acknowledgments
About the Author
An Introduction to Part I: Preliminary Issues
	Chapter 1 Introduction and Overview
		1.1.1 A Comparative Introduction, Part I
		1.1.2 A Comparative Introduction, Part II
		1.2 You Can “Pick and Choose” How You Read This Book
		1.3 The Layered Nature of the Mathematics in ESA
		1.4 An Overview
	Chapter 2 Polygons and Stars
		2.1 How Polygons Are Drawn in This Book
		2.2.1 Stars That Work, and Stars That Don’t
		2.2.2 How Many Points Does a Continuously-Drawn n,J-star Have?
		2.3.1 Sharpest Stars
		2.3.2 How Sharpest Stars Are Drawn
		2.3.3 Not All Even Stars Are Created Equal
		2.4.1 Angles from Regular Polygons and Stars
		2.4.2 MA. Two Questions About Angles
		2.4.3 MA. A Totally Optional Addition to Section 2.4
		2.4.4 On Why the Angles Table Has a Ragged but Regular, Right Edge Pattern
		2.5.1 Viewing Stars as Rotating Polygons
		2.5.2 MA. What Is Regular About the Angles in Rotating Polygons and Stars?
		2.6.1 Stars Inside of a Star
		2.6.2 MA. Analyzing Stars Inside of a Star
		2.6.3 Calculating Triangle Angles Using Vertices
		2.7.1 Challenge Questions for Chapter 2: Analyzing Jumps, J
		2.7.2 Stars as Rotating Polygons Challenge Questions
		2.7.3 Creating Internal 12,3 and 12,4-Stars Challenge Questions
		2.7.4 Triangle Angles Challenge Questions
An Introduction to Part II: String Art
	Chapter 3 String Art Basics
		3.1 A Primer on S, the Number of Subdivisions between Vertices
		3.2 A Primer on P, the Number of Subdivisions Between Points
		3.3 How Lines Are Created from S and P
	Chapter 4 Issues Involving Commonality
		4.1 Finding the Total Number of Connected Line Segments in an Image, VCF and SCF
		4.2.1 Visualizing the Vertex Frame: The Visible Number of Line Segments Might Differ from the Calculated Number of Segments
		4.2.2 Drawing the Vertex Frame
		4.2.3 MA. Analyzing Patterns in Continuously-Drawn n,J-Stars
		4.2.4 An Analytical Approach to Line Placement
		4.3 Symmetry About n·S/2 and the Number of Distinct Images as a Function of P for Given Values of n, S, and J
	Chapter 5 Cycles
		5.1 The Number of Segments in a Cycle and the Number of Cycles in a Circuit
		5.2 Where the First Cycle Ends Tells Us How the Image Is Filled In
		5.3 Using Cycles to Understand Images
		5.4.1 Image Density: The Role of Vertex Common Factor, VCF = GCD(n,J)
		5.4.2 Image Density: The Role of Subdivision Common Factor, SCF = GCD(n·S/VCF,P)
		5.4.3 MA. How Image Density Is Distributed Between Concentric Circles of Internal Subdivision Points and Polygonal Vertices
		5.5 Single-Cycle Images
	Chapter 6 Alternative ways to Obtain an Image
		6.1 On Finding the Smallest Values of n, S, P, and J that Create an Image
		6.2 There Are Only Two Ways to Draw an Image (but there are four ways to describe it using (n, S, P, J))
		6.3.1 A Primer on Vertical Symmetry
		6.3.2 A Slick Way to Verify Symmetry of J and n-J (using Excel file, 10.0.1)
		6.4.1 Rotational Symmetry Versus Lines of Symmetry
		6.4.2 Lines of Symmetry, Take 2
	Chapter 7 Levels of Subdivision Points
		7.1 Subdivisions Create Concentric Circles of Possible Points
		7.2 One Level Change Images
		7.3 Level Patterns Across a Cycle
		7.4 Symmetry Across a Cycle
	Chapter 8 Shape-Shifting Polygons
		8.1 Shape-Shifting Polygons
		8.2 Role of J in Shape-Shifting Polygons:
		8.3.1 Switching S and n, Take 1
		8.3.2 Switching S and n, Take 2
		8.4 Three Shape-Shifting Triangles
		8.5.1 Single-Step Images
		8.5.2 Composite Squares Produce Multiple Single-Step Polygons and Polygrams
		8.6 Kicking the Tires of Three Shape-Shifting Triangles
	Chapter 9 An Overarching Question
		9.1 Searching for Similarity: Exploring the Star-in-a-Star Pattern
		9.2.1 Reverse-Engineering an Image
		9.2.2 Additional Image Detective Strategies
		9.3 Curves from Lines and Points
		9.4 Smallest-Step Images
		9.5 Comparing Single-Step with Smallest-Step Using Three Shape-Shifting Triangles
		9.6 Pushing the Bounds of an Image Type
		9.7 Using Shape-Shifting Stars to Explore Curves from Lines
	Chapter 10 Functionally Modified String Art Files
		10.1 Functionally Enabling n, S, P, and J
		10.2.1 Analyzing Waves of Images and the Subdivision Donut Hole
		10.2.2 MA. Automating P to Produce Images That Avoid the Donut Hole
		10.2.3 Create Your Own Connections: Exploring Modified Brunes Stars
		10.3 Partial-Way-Around Images: Systematically Encroaching on the Donut Hole
		10.4 Pulsing Images
		10.5 MA. Polygons and Stars in a Cycle
	Chapter 11 A Sampling of Image Archetypes
		11.1 Some Examples of String Art Image Archetypes
		11.2 Curved-Tip Stars
		11.3 Porcupine Polygons
		11.4 Porcupine Stars Versus Polygons
		11.5.1 Sunbursts
		11.5.2 Variations on Sunbursts: Increasing S Beyond 2 While Maintaining Polygonal Status
		11.6.1 Quivering Polygons
		11.6.2 The Role of J in Quivering Polygonal Images
		11.6.3 The Relative Size of S and P in Quivering Polygons
		11.6.4 MA. Why Quivering Polygon Peaks Rotate the Way They Do (a Triangular Example)
		11.7 Odd Needle Stars
		11.8.1 Spinning Needle Stars
		11.8.2 Spinning Needle Stars, Take 2: Even and Odd Dancing Partners
		11.9 Ultra-Needles and the Benefit of Single Line Drawing Mode
		11.10.1 Stacked Circles
		11.10.2 Stacked Circles With a Focus on Levels
		11.10.3 Stacked Circles, Take 3: Variations on the Theme
		11.11 Chrysanthemums
		11.12 Clothespins as Spinning Needle Stars Turned Inside-Out
	Chapter 12 n = P Images
		12.1 An Introduction to n = P Images
		12.2 n = P Image Archetypes
		12.3 Divisibility and the Elusive 6,2-Star
		12.4 Small Images
		12.5 Counting Strands and Loops in Coiled Rope Images
		12.6 Two Footballs
		12.7 MA. Modular Analysis of Two-Football Cycles
		12.8 A Quivering Triangle That Is Actually Shape-Shifting
		12.9 Finger Traps
		12.10 On the Distance Between Paired Suspension Curves
		12.11 Generalized Stars: Exploring One-off Images Using 7-point Stars (and More)
		12.12 n = P Porcupines
	Chapter 13 60-Second Images
		13.1 60-Second Images
		13.2 60 Polygon or Polygram Images (an Alternative to 60-Second Images)
		13.3 Alternative Universes
	Chapter 14 Challenge Questions for Part II
		14.1 Introductory Challenge Questions
		14.2 Octagonal Challenge Questions
		14.3 Areas of n = 4 Images
		14.4 Brunes Star Questions
		14.5 Image Detective (Using Sections 4.1 and 9.2.1)
		14.6 Use a Ruler to Analyze These Image Detective Questions (Based on Section 9.2.1)
		14.7 Two Footballs Challenge Questions
		14.8 Searching for Squares Inside a Modified Brunes Star
		14.9 Calculating Areas of Squares in Modified Brunes Stars
		14.10 Can you find Similar Images?
An Introduction to Part III: Variations on the String Art Model
	Chapter 15 Centered-Point Flowers
		15.1 Understanding the Vertex Jump Pattern for Centered-Point Flowers, CPF
		15.2 Finding the Total Number of Connected Line Segments in Centered-Point Flowers
		15.3.1 Subdivision Patterns and What This Implies About S and P for Centered-Point Flowers
		15.3.2 P/S Helps You Find Similar Images
		15.4 Why Do Some Subdivision Endpoints Have More Lines Than Others?
		15.5.1 Centered-Point Flower Images When P Is a Multiple of S
		15.5.2 P = 2S Images: Even and Odd Spikes
		15.5.3 A Deeper Dive into Spiked Images
		15.6 Creating Functionally Related Images: A P(S) Example
		15.7.1 SCF = 1 Images
		15.7.2 SCF > 1 Images
	Chapter 16 Double Jump Models
		16.1 Calculating Lines Used Given Jump Sets
		16.2 An Excel Primer on Double Jump Models (While Achieving the Elusive 6,2-Star)
		16.3 A Visual Interpretation of VCF, SCF, and Image Density
		16.4 Revisiting Single-Step Images (from Section 8.5.1)
		16.5 Functionally Related Double Jump Models
		16.6 What to Do When You Find a Nice Image
		16.7 Needle Fans and Blade Fans
		16.8 A Sampling of Images from the Double Jump Model
		16.9 Pentagonal People
	Chapter 17 Four-Color Clock Arithmetic
		17.1 An Introduction to Triple Jump Models, VCF = 1
		17.2 Triple Jumps with VCF > 1
		17.3 Spikes and Tails
		17.4.1 Comparing 1, 2, and 3 Jump Set 12,5-Stars
		17.4.2 A Line Analysis of a J = 0 Image
		17.5 Venturing Beyond Curved-Tip Stars
	Chapter 18 Larger Jump Set Models
		18.1 An Introduction to Larger Jump Set Models
		18.2 Mystic Roses
		18.3 Counting Backwards
		18.4.1 From VF to String Art with Larger Jump Sets
		18.4.2 Just Tails, Asymmetric Suspension Curves, and Spirals
		18.4.3 Fibonacci
		18.5 Creating Videos
	Chapter 19 Busting Out of Our Polygonal Constraint
		19.1 What Changes Once We Leave Our Regular Polygonal World? The Role of V
		19.2 Linked Vertices
		19.3 Listing Vertices Twice in a Row (Recreating a Zero Jump)
		19.4 The Four-Color Model, Exploring Inside the Box
		19.5 Moving Beyond Inside the Box (Section 19.4) and Letting the Image Swirl
		19.6 Creating Your Initials in Four Colors
	Chapter 20 Challenge Questions for Part III
		20.1 CPF Image Analysis
		20.2 CPF Area Analysis
		20.3 CPF Point Location Challenge Questions
		20.4 Comparing Images Across Chapters
		20.5 Area of Vertex Frame in the Double Jump Model
		20.6 Double Jump “Hours” Challenge Questions
		20.7 Four-Color Challenge Questions, Part I, Vertex Frame Analysis
		20.8 Four-Color Challenge Questions, Part II, Comparing Jump Sets
		20.9 Four-Color Challenge Questions, Part III, Color Density Questions
		20.10 Four-Color Challenge Questions, Part IV, Image Analysis
		20.11 Really Sharp Triangles, Part I
		20.12 Really Sharp Triangles, Part II
		20.13 Fibonacci Golden Spiral Approximation in Excel (A Guided Challenge Question)
		20.14 Bird Beak Challenge Questions
		20.15 Variations on Being Inside the Box
		20.16 Four-Point Stars
An Introduction to Part IV: Issues, Mathematical and Otherwise
	Chapter 21 Basic Properties of Numbers
		21.1.1 About Prime and Composite Numbers
		21.1.2 Identifying Prime Numbers Using the Ancient Sieve of Eratosthenes (From the 3rd Century BC)
		21.2.1 Commonality Between Numbers
		21.2.2 About Relatively Prime (or Coprime) Numbers
		21.2.3 MA. Euclid’s Algorithm
		21.3 What is the Product of Two Numbers that Differ by an Even Amount? (This looks at times table patterns along diagonals.)
	Chapter 22 Angles in Polygons and Stars
		22.1.1 Inscribed Angles and Central Angles
		22.1.2 Why the Inscribed Angle Theorem Works
		22.2 About Interior Angles and Parallel Lines
		22.3 About Implied Exterior Angles
	Chapter 23 Modular Arithmetic
		23.1 About MOD: When Counting Round and Round, the Remainder Is All That Matters
		23.2 MA. MOD Take 2: Additional Properties of Modular Arithmetic
		23.3 MA. Counting Backwards (Modular Arithmetic, Take 3)
		23.4 MA. A(nother) Party Trick
	Chapter 24 Modular Multiplicative Inverses, MMI
		24.1 Introduction to Modular Multiplicative Inverses
		24.2 MA. MMI and Negative MMI (nMMI)
		24.3 MA. Backtracking Euclid’s Algorithm to Find Modular Multiplicative Inverse
	Chapter 25 A Guide to the Web Model
		25.1 An Introduction to the Web Version of String Art
		25.2 Changing Numbers in the Web Version
		25.3 How Jumps are Incorporated into the Web Version of String Art
		25.4 Using the Web Version Drawing Mode
		25.5.1 A Guided Inquiry to the Web Version for Younger Users: Entering Numbers
		25.5.2 A Guided Inquiry to the Web Version for Younger Users: Web Jumps
		25.5.3 A Guided Inquiry to the Web Version for Younger Users: Drawing Modes
	Chapter 26 Suggestions for Mathematics Teachers
		26.1 Introductory Remarks for Teachers
		26.2 A Quick-Start Guide to Using the Polygons and Stars for Teachers Excel File 2.0.2
		26.3 A Quick-Start Guide to Using the String Art for Teachers Excel File 3.0.3
		26.4 A Teaching Companion to the Web Guide for Younger Users
		26.5 Making Your Own Handouts Using the Excel Files
	Chapter 27 Challenge Question Answers
	Chapter 28 Glossary of Commonly Used Terms and Abbreviations and Index
Bibliography