Kitchen Science Fractals: A Lab Manual for Fractal Geometry

Contents
Prologue
Software and solutions
1 IFS Labs
1.1 Finding IFS for fractal images
1.1.1 Purpose
1.1.2 Materials
1.1.3 Background
1.1.3.1 Geometry of plane transformations
1.1.3.2 IFS formalism
1.1.3.3 The deterministic algorithm
1.1.3.4 The inverse problem
1.1.4 Procedure
1.1.5 Sample A
1.1.6 Sample B
1.1.7 Conclusion
1.1.8 Exercises
1.2 Spiral fractals from IFS
1.2.1 Purpose
1.2.2 Materials
1.2.3 Background
1.2.3.1 Decomposition of a spiral
1.2.3.2 The random algorithm
1.2.3.3 Computing the probabilities
1.2.3.4 The effect of scaling the translation terms
1.2.4 Procedure
1.2.5 Sample
1.2.6 Conclusion
1.2.7 Exercises
1.3 Finding IFS rules from images of points
1.3.1 Purpose
1.3.2 Materials
1.3.3 Background
1.3.3.1 General formulation
1.3.3.2 Matrix formulation
1.3.3.3 Proof of a unique solution
1.3.3.4 And the solution is
1.3.3.5 Conversion to r, S, θ, and φ
1.3.4 Sample
1.3.5 Conclusion
1.3.6 Exercises
1.4 A fractal leaf by IFS
1.4.1 Purpose
1.4.2 Materials
1.4.3 Background
1.4.4 Procedure
1.4.5 Sample
1.4.6 Conclusion
1.4.7 Exercises
1.5 Fractal wallpaper
1.5.1 Purpose
1.5.2 Materials
1.5.3 Background
1.5.4 Procedure
1.5.5 Sample
1.5.6 Conclusion
1.5.7 Exercises
1.6 Cumulative gasket pictures
1.6.1 Purpose
1.6.2 Materials
1.6.3 Background
1.6.3.1 The chaos game
1.6.3.2 The random IFS algorithm
1.6.3.3 The chaos game as random IFS
1.6.3.4 A shortcoming of the chaos game played manually
1.6.4 Procedure
1.6.5 Sample
1.6.6 Conclusion
1.6.7 Exercises
1.7 IFS and addresses
1.7.1 Purpose
1.7.2 Materials
1.7.3 Background
1.7.4 Sample
1.7.5 Conclusion
1.7.6 Exercises
1.8 Decimals as addresses
1.8.1 Purpose
1.8.2 Materials
1.8.3 Background
1.8.4 Procedure
1.8.5 Sample A
1.8.6 Sample B
1.8.7 Conclusion
1.8.8 Exercises
1.9 IFS with memory
1.9.1 Purpose
1.9.2 Materials
1.9.3 Background
1.9.3.1 Allowed pairs
1.9.3.2 Lines
1.9.3.3 Romes
1.9.4 Procedure
1.9.5 Sample 1
1.9.6 Sample 2
1.9.7 Conclusion
1.9.8 Exercises
1.10 IFS with more memory
1.10.1 Purpose
1.10.2 Materials
1.10.3 Background
1.10.4 Procedure
1.10.5 Sample A
1.10.6 Sample B
1.10.7 Conclusion
1.10.8 Exercises
1.11 Data analysis by driven IFS
1.11.1 Purpose
1.11.2 Materials
1.11.3 Background
1.11.3.1 The driven IFS algorithm
1.11.3.2 IFS driven by symbol sequences
1.11.3.3 Numerical data binning schemes
1.11.3.4 Interpretation of driven IFS plots
1.11.3.5 Markov partitions of data
1.11.3.6 IFS driven by iteration
1.11.4 Procedure
1.11.5 Sample
1.11.6 Conclusion
1.11.7 Exercises
2 Dimension and Measurement Labs
2.1 Dimension by box-counting
2.1.1 Purpose
2.1.2 Materials
2.1.3 Background
2.1.4 Procedure
2.1.5 Sample A: The dimension of Grenada Lake
2.1.6 Sample B: Measuring in dimensions right and wrong
2.1.7 Sample C: The dimension of a product
2.1.8 Conclusion
2.1.9 Exercises
2.1.10 Grids
2.2 Paper ball and bean bag dimensions
2.2.1 Purpose
2.2.2 Materials
2.2.3 Background
2.2.3.1 Mass dimension
2.2.3.2 Dimensions and intersections
2.2.4 Procedure
2.2.5 Sample A. Crumpled paper balls
2.2.6 Sample B
2.2.7 Conclusion
2.2.8 Exercises
2.3 Calculating similarity dimension
2.3.1 Purpose
2.3.2 Materials
2.3.3 Background
2.3.4 Procedure
2.3.5 Sample A
2.3.6 Sample B
2.3.7 Conclusion
2.3.8 Exercises
2.4 Sierpinski tetrahedron
2.4.1 Purpose
2.4.2 Materials
2.4.3 Background
2.4.3.1 The tetrahedron
2.4.3.2 The Sierpinski tetrahedron
2.4.3.3 The Sierpinski tetrahedron complement
2.4.4 Procedure
2.4.5 Sample
2.4.6 Conclusion
2.4.7 Exercises
2.5 Koch tetrahedron
2.5.1 Purpose
2.5.2 Materials
2.5.3 Background
2.5.4 Procedure
2.5.5 Sample
2.5.6 Conclusion
2.5.7 Exercises
2.5.8 Templates
2.6 Sierpinski hypertetrahedron
2.6.1 Purpose
2.6.2 Materials
2.6.3 Background
2.6.4 Procedure
2.6.5 Sample A
2.6.6 Sample B
2.6.7 Conclusion
2.6.8 Exercises
2.7 Basic multifractals: f(α) curves
2.7.1 Purpose
2.7.2 Materials
2.7.3 Background
2.7.3.1 The generalized Moran equation
2.7.3.2 The Hölder exponent
2.7.3.3 The f(α) curve
2.7.4 Procedure
2.7.5 Sample A
2.7.6 Sample B
2.7.7 Sample C
2.7.8 Sample D
2.7.9 Conclusion
2.7.10 Exercises
3 Iteration Labs
3.1 Visualizing iteration patterns
3.1.1 Purpose
3.1.2 Materials
3.1.3 Background
3.1.3.1 Graphical iteration
3.1.3.2 Time series plots
3.1.3.3 Occupancy histograms
3.1.3.4 Return maps
3.1.3.5 Kelly plots
3.1.4 Procedure
3.1.5 Sample A
3.1.6 Sample B
3.1.7 Conclusion
3.1.8 Exercises
3.2 Synchronized chaos
3.2.1 Purpose
3.2.2 Materials
3.2.3 Background
3.2.3.1 Two maps
3.2.3.2 Three maps
3.2.3.3 A fractal network
3.2.4 Procedure
3.2.5 Sample A
3.2.6 Sample B
3.2.7 Conclusion
3.2.8 Exercises
3.3 Domains of compositions
3.3.1 Purpose
3.3.2 Materials
3.3.3 Background
3.3.4 Procedure
3.3.5 Sample
3.3.6 Conclusion
3.3.7 Exercises
3.4 Fractals and Pascal’s triangles
3.4.1 Purpose
3.4.2 Materials
3.4.3 Background
3.4.3.1 Pascal’s triangle and binomial coefficients
3.4.3.2 Pascal’s triangle and the Sierpinski gasket
3.4.3.3 Pascal’s triangle and other number patterns
3.4.3.4 Some group theory
3.4.4 Procedure
3.4.5 Sample A
3.4.6 Sample B
3.4.7 Conclusion
3.4.8 Exercises
3.5 Fractals and Pascal’s triangle relatives
3.5.1 Purpose
3.5.2 Materials
3.5.3 Background
3.5.3.1 A tiny bit more group theory
3.5.3.2 Symmetries of regular polygons
3.5.3.3 Group tables
3.5.3.4 Permutations
3.5.4 Procedure
3.5.5 Sample A
3.5.6 Sample B
3.5.7 Conclusion
3.5.8 Exercises
3.6 Mandelbrot sets and Julia sets
3.6.1 Purpose
3.6.2 Materials
3.6.3 Background
3.6.4 Procedure
3.6.5 Sample A
3.6.6 Sample B
3.6.7 Sample C
3.6.8 Conclusion
3.6.9 Exercises
3.7 Circle inversion fractals
3.7.1 Purpose
3.7.2 Materials
3.7.3 Background
3.7.3.1 Circle inversion definition
3.7.3.2 Circle inversion properties
3.7.3.3 Limit sets
3.7.3.4 Circle inversion limit sets
3.7.3.5 Some circle inversion limit set examples
3.7.4 Procedure
3.7.5 Sample A
3.7.6 Conclusion
3.7.7 Exercises
3.8 Fractal tiles
3.8.1 Purpose
3.8.2 Materials
3.8.3 Background
3.8.3.1 Opposite side modification
3.8.3.2 Some matrix arithmetic
3.8.3.3 Matrices and fractal tiles
3.8.4 Procedure
3.8.5 Sample A
3.8.6 Sample B
3.8.7 Sample C
3.8.8 Conclusion
3.8.9 Exercises
4 Labs in the Studio and in the Kitchen
4.1 Fractal painting: decalcomania 1
4.1.1 Purpose
4.1.2 Materials
4.1.3 Background
4.1.3.1 Decalcomania
4.1.3.2 Branch mechanics
4.1.3.3 A brief review of box-counting dimension
4.1.3.4 Number of branches
4.1.3.5 Number of levels of branching
4.1.3.6 Ratios of lengths of successive branches
4.1.4 Procedure
4.1.5 Sample A. Decalcomania variations.
4.1.6 Sample B. Measurement experiments.
4.1.7 Conclusion
4.1.8 Exercises
4.2 Fractal painting: decalcomania 2
4.2.1 Purpose
4.2.2 Materials
4.2.3 Background
4.2.4 Procedure
4.2.5 Sample A. Spreading a blob of paint by pressure.
4.2.6 Sample B. Applying paint with a brush.
4.2.7 Conclusion
4.2.8 Exercises
4.3 Fractal painting: bleeds
4.3.1 Purpose
4.3.2 Materials
4.3.3 Background
4.3.4 Procedure
4.3.5 Sample A. Central acrylic bleeds
4.3.6 Sample B. Edge bleeds
4.3.7 Sample C. Watercolor bleeds
4.3.8 Conclusion
4.3.9 Exercises
4.4 Fractal painting: mixing
4.4.1 Purpose
4.4.2 Materials
4.4.3 Background
4.4.4 Procedure
4.4.5 Sample A
4.4.6 Sample B
4.4.7 Conclusion
4.4.8 Exercises
4.5 Fractal painting: dripping
4.5.1 Purpose
4.5.2 Materials
4.5.3 Background
4.5.4 Procedure
4.5.5 Sample A
4.5.6 Sample B
4.5.8 Exercises
4.6 Fractal paper folds
4.6.1 Purpose
4.6.2 Materials
4.6.3 Background
4.6.4 Procedure
4.6.5 Sample A
4.6.6 Sample B
4.6.7 Gallery of examples
4.6.8 Conclusion
4.6.9 Exercises
4.7 A closer look at leaves
4.7.1 Purpose
4.7.2 Materials
4.7.3 Background
4.7.4 Procedure
4.7.5 Sample
4.7.6 Conclusion
4.7.7 Exercises
4.8 Structures of vegetables
4.8.1 Purpose
4.8.2 Materials
4.8.3 Background
4.8.4 Procedure
4.8.5 Sample A. Broccoli Romanesco
4.8.6 Sample B. Electrical tape and kale
4.8.7 Sample C. Sedum
4.8.8 Conclusion
4.8.9 Exercises
4.9 Cooking fractals
4.9.1 Purpose
4.9.2 Materials
4.9.3 Background
4.9.4 Procedure
4.9.5 Sample A. Crepes
4.9.6 Sample B. Bread and butter, and beer
4.9.7 Conclusion
4.9.8 Exercises
5 Labs in the Lab
5.1 Magnetic pendulum
5.1.1 Purpose
5.1.2 Materials
5.1.3 Background
5.1.3.1 Basins of attraction
5.1.3.2 Newton’s method
5.1.3.3 Basin boundaries
5.1.3.4 Fractal basin boundaries
5.1.3.5 The Wada property
5.1.4 Procedure
5.1.5 Sample A
5.1.6 Sample B
5.1.7 Conclusion
5.1.8 Exercises
5.2 Optical gasket
5.2.1 Purpose
5.2.2 Materials
5.2.3 Background
5.2.4 Procedure
5.2.5 Sample A
5.2.6 Sample B
5.2.7 Conclusion
5.2.8 Exercises
5.3 Video feedback fractals
5.3.1 Purpose
5.3.2 Materials
5.3.3 Background
5.3.4 Procedure
5.3.5 Sample A
5.3.6 Sample B
5.3.7 Conclusion
5.3.8 Exercises
5.4 Electrodeposition
5.4.1 Purpose
5.4.2 Materials
5.4.3 Background
5.4.4 Procedure
5.4.5 Sample A
5.4.6 Sample B
5.4.7 Sample C
5.4.8 Conclusion
5.4.9 Exercises
5.5 Viscous fingering
5.5.1 Purpose
5.5.2 Materials
5.5.3 Background
5.5.4 Procedure
5.5.5 Sample A: Glycerine experiments
5.5.6 Sample B: Guar gum solution experiments
5.5.7 Conclusion
5.5.8 Exercises
5.6 Crumpled paper patterns
5.6.1 Purpose
5.6.2 Materials
5.6.3 Background
5.6.4 Procedure
5.6.5 Sample
5.6.6 Conclusion
5.6.7 Exercises
5.7 Fractal networks of resistors
5.7.1 Purpose
5.7.2 Materials
5.7.3 Background
5.7.4 Procedure
5.7.5 Sample A. Measurements of RAB(n).
5.7.6 Sample B. The effect of breaking a connection.
5.7.7 Sample C. Measurement of the resistance between different points.
5.7.8 Conclusion
5.7.9 Exercises
5.8 Fractal networks of magnets
5.8.1 Purpose
5.8.2 Materials
5.8.3 Background
5.8.4 Procedure
5.8.6 Sample B. Sierpinski gasket patterns
5.8.7 Conclusion
5.8.8 Exercises
5.9 Synchronization in fractal networks of oscillators
5.9.1 Purpose
5.9.2 Materials
5.9.3 Background
5.9.4 Procedure
5.9.5 Sample A. One oscillator.
5.9.6 Sample B. Two oscillators.
5.9.7 Sample C. Three oscillators.
5.9.8 Sample D. Nine oscillators.
5.9.9 Conclusion
5.9.10 Exercises
6 What Else?
6.1 Building block fractals
6.2 Non-Euclidean tilings
6.3 Fractal perimeters
6.4 Multifractal finance
6.5 Fractal music
6.6 Other ideas
6.6.1 Lichtenberg figures
6.6.2 Fractality of a life
6.6.3 Fractals in literature
6.6.4 Fractal government
6.6.5 Found fractals: a photography assignment
6.6.6 Fractal dance
6.6.7 A fractal play
7 Why labs matter
A Specific Physical Supplies
B Technical Notes
B.1 Notes for finding IFS, Lab 1.1
B.1.1 Compact sets
B.1.2 Hausdorff distance
B.1.3 Convergence of the deterministic algorithm
B.2 Notes for spiral fractals, Lab 1.2
B.2.1 Fixed points and IFS
B.2.2 Comparing the random and deterministic algorithms
B.3 Notes for cumulative gasket pictures, Lab 1.6
B.4 Notes for IFS with more memory, Lab 1.10
B.5 Notes on entropy and partitions, Lab 1.11.
B.6 Notes on linear regression, Lab 2.1
B.7 Notes on the algebra of dimensions, Labs 2.1 and 2.2.
B.8 Notes on eigenvalues and the Moran equation, Lab 2.3.
B.8.1 Eigenvalues and eigenvectors
B.8.2 Moran equation solutions
B.9 Notes on multifractal analysis, Lab 2.7
B.9.1 Hölder exponents
B.9.2 Concavity of the f(α) curve
B.9.3 Attractor dimension and the f(α) curve
B.9.4 A formula for the f(α) curve
B.10 Notes on the Mandelbrot set and Juliasets, Lab 3.6
B.10.1 Escape to infinity criteria
B.10.2 Some Mandelbrot set geometry
B.10.3 Simplified polynomials
B.11 Notes on circle inversion fractals, Lab 3.7
B.11.1 Circle inversion properties
B.11.2 Limit set examples
B.12 Notes on fractal painting: dripping, Lab 4.5
B.13 Notes on power law measurements, Lab 4.9
B.14 Notes on magnetic pendulum differential equations, Lab 5.1
B.15 Notes on molarity calculations, Lab 5.4
B.16 Notes on fractal resistor networks Lab 5.7
B.17 Notes on synchronization in fractal networks of oscillators, Lab 5.9
References
Prologue
1 IFS labs
1.1 Finding IFS for fractal images
1.5 Fractal wallpaper
1.9 IFS with memory
1.10 IFS with more memory
1.11 Data analysis by driven IFS
2.1 Dimension by box-counting
2.2 Paper ball and bean bag dimensions
2.3 Calculating similarity dimension
2.5 Koch tetrahedron
2.6 Sierpinski hypertetrahedron
2.7 Basic multifractals
3.1 Visualzing iteration patterns
3.2 Synchronized chaos
3.3 Domains of compositions
3.4 Fractals and Pascal’s triangle
3.6 Mandelbrot sets and Julia sets
3.7 Circle inversion fractals
3.8 Fractal tiles
4.1 Fractal painting: decalcomania 1
4.2 Fractal painting: decalcomania 2
4.3 Fractal painting: bleeds
4.4 Fractal painting: mixing
4.5 Fractal painting: dripping
4.6 Fractal folds
4.7 A closer look at leaves
4.9 Cooking fractals
5.1 Magnetic pendulum
5.2 Optical gasket
5.3 video feedback fractals
5.4 Electrodeposition
5.5 Viscous fingering
5.6 Crumpled paper patterns
5.7 Fractal resistor networks
5.8 Fractal networks of magnets
5.9 Fractal network synchronization
6.2 Non-Euclidean tessellations
6.4 Multifractal finance
6.5 Fractal music
6.6.1 Lichtenberg figures
6.6.3 Fractals in literature
6.6.6 Fractal dance
B.17 Acknowledgments
Figure Credits
Acknowledgements
Index