Contents -- Introduction -- Acknowledgements -- Permission -- Part I Essays in Geometry and Number as They Arise in Nature Music Architecture and Design -- Chapter 1 The Spiral in Nature and Myth -- 1.1 Introduction -- 1.2 The Australian Aborigines -- 1.3 The Fali -- 1.4 The Precession of the Equinoxes in Astronomy and Myth -- 1.5 Spiral Forms in Water -- 1.6 Meanders -- 1.7 Wave Movement -- 1.8 Vortices and Vortex Trains -- 1.9 Vortex Rings -- 1.10 Three Characteristic Features of Water -- 1.11 The Flowform Method -- 1.12 Conclusion -- Chapter 2 The Vortex of Life -- 2.1 Introduction -- 2.2 Projective Geometry -- 2.3 Perspective Transformations on the Line to Points on a Line -- 2.4 Projective Transformations of Points on a Line to Points on a Line -- 2.5 Growth Measures -- 2.6 Involutions -- 2.7 Circling Measures -- 2.8 Path Curves -- 2.9 Path Curves in Three Dimensions -- 2.10 Field of Form -- 2.11 Comparison of Three Systems -- 2.12 Conclusion -- Appendix 2.A Homogeneous Coordinates -- Chapter 3 Harmonic Law -- 3.1 Introduction -- 3.2 Musical Roots of Ancient Sumeria -- 3.3 Musical Fundamentals -- 3.4 Spiral Fifths -- 3.5 Just Tuning -- 3.6 Music and Myth -- 3.7 Musically Encoded Dialogues of Plato -- 3.8 The Mathematical Structure of the Tonal Matrix -- 3.9 The Color Wheel -- 3.10 Conclusion -- Appendix 3.A -- 3.A.1 Logarithms and the Logarithmic Spiral -- 3.A.2 Properties of Logarithms -- 3.A.3 Logarithms and the Musical Scale -- Appendix 3.B The Pythagorean Comma -- Appendix 3.C Vectors -- Chapter 4 The Projective Nature of the Musical Scale -- 4.1 Introduction -- 4.2 A Perspective View of the Tonal Matrix: The Overtone Series -- 4.3 The Three Means -- 4.4 Projective Analysis of an Egyptian Tablet -- 4.4.1 An Analysis of Schwaller De Lubicz's Number Sequence -- 4.5 Conclusion -- Appendix 4.A.

Chapter 5 The Music of the Spheres -- 5.1 Introduction -- 5.2 The Music of the Spheres -- 5.3 Kepler's Music of the Spheres -- 5.4 Results of Kepler's Analysis -- 5.5 Bode's Law -- 5.6 A Musical Relationship that Kepler Overlooked -- 5.7 Conclusion -- Appendix 5.A Kepler's Ratios -- Chapter 6 Tangrams and Amish Quilts -- 6.1 Introduction -- 6.2 Tangrams -- 6.3 Amish Quilts -- 6.4 Zonogons -- 6.5 Zonohedra -- 6.6 N-Dimensional Cubes -- 6.7 Triangular Grids in Design: An Islamic Quilt Pattern -- 6.8 Other Zonogons -- 6.9 Conclusion -- Appendix 6.A -- 6.A.1 Steps to creating a triangular grid of circles -- 6.A.2 Steps to creating a square circle grid -- Chapter 7 Linking Proportions Architecture and Music -- 7.1 Introduction -- 7.2 The Musical Proportions of the Italian Renaissance -- 7.3 The Roman System of Proportions -- 7.4 The Geometry of the Roman System of Proportions -- 7.5 The Law of Repetition of Ratios -- 7.6 Relationship between the Roman System and the System of Musical Proportions -- 7.7 Ehrenkrantz' System of Modular Coordination -- 7.8 Conclusion -- Appendix 7.A An Ancient Babylonian Method for Finding the Square Root of 2 -- Chapter 8 A Secret of Ancient Geometry -- 8.1 Introduction -- 8.2 The Concept of Measure in Ancient Architecture -- 8.3 The Ancient Geometry of Tons Brunes -- 8.4 Equipartion of Lengths: A Study in Perspective -- 8.5 The 3 4 5-Triangle in Sacred Geometry and Architecture -- 8.5.1 Construction of the Brunes star from 3 4 5-triangles -- 8.5.2 The 3 4 5-triangle and its musical proportions -- 8.5.3 The geometry of the Brunes star -- 8.6 What Pleases the Ear Should Please the Eye -- 8.7 Conclusion -- Appendix 8.A Harmonic Means -- Appendix 8.B Projective Analysis of the Equipartition Properties of the Brunes Star -- Chapter 9 The Hyperbolic Brunes Star -- 9.1 Introduction.

9.2 A Generalized Brunes Star -- 9.3 Zeno's Hyperbolic Paradox -- 9.4 Hyperbolic Functions and Number -- 9.5 Hyperbolic Functions in the Theory of Probability -- 9.6 Gambler's Ruin -- 9.7 Little End of the Stick Problem -- 9.8 Shannon's Entropy Function and Optimal Betting Strategy -- 9.9 The Generalized Little End of the Stick Problem -- 9.10 Conclusion -- Chapter 10 The Hidden Pavements of the Laurentian Library -- 10.1 Introduction -- 10.2 The Laurentian Library -- 10.3 Reconstruction of the Pavements -- 10.4 The Sacred-Cut Panel -- 10.5 The Medici Panel -- 10.6 The Mask Panel -- 10.7 Conclusion -- Appendix 10.A The Sacred Cut and the Square Circle Grid -- Chapter 11 Measure in Megalithic Britain -- 11.1 Introduction -- 11.2 A Standard Measure -- 11.3 Megalithic British and Greek Measures Compared -- 11.4 Statistical Studies of Megalithic Measure -- 11.5 Measurements at Mid Clyth -- 11.6 The Stone Circles -- 11.7 Flattened Circles and the Golden Mean -- 11.8 Historical Perspective -- 11.9 Conclusion -- Appendix 11. A -- Appendix 11. B The Geometry of the Stone Circles -- Chapter 12 The Flame-hand Letters of the Hebrew Alphabet -- 12.1 Introduction -- 12.2 The Flame-Hand Letters of the Hebrew Alphabet -- 12.3 The Vortex Defining the Living Fruit -- 12.4 The Torus -- 12.5 The Tetrahelix -- 12.6 The Meaning of the Letters -- 12.6.1 Oneness -- 12.6.2 Distinction -- 12.7 Generation of the Flame-Hand Letters -- 12.8 Some Commentary on Tenen's Proposal -- 12.9 Conclusion -- Part II Concepts Described in Part I Reappear in the Context of Fractals Chaos Plant Growth and Other Dynamical Systems -- Chapter 13 Self-Referential Systems -- 13.1 Introduction -- 13.2 Self-Referential Systems in Mathematics -- 13.3 The Nature of Self-Referentiality.

13.4 Self-Referentiality and the Egyptian Creation Myth -- 13.5 Spencer-Brown's Concept of Re-entry -- 13.6 Imaginary Numbers and Self-Referential Logic -- 13.7 Knots and Self-Referential Logic -- 13.8 Conclusion -- Appendix 13.A -- Chapter 14 Nature's Number System -- 14.1 Introduction -- 14.2 The Nature of Rational and Irrational Numbers -- 14.3 Number -- 14.4 Farey Series and Continued Fractions -- 14.5 Continued Fractions Gears Logic and Design -- 14.6 Farey Series and Natural Vibrations -- 14.7 Conclusion -- Appendix 14.A Euler's y-Function -- Appendix 14.B The Relation between Continued Fraction Indices and the Little End of the Stick Problem -- Appendix 14.C "Kissing" Gears -- Chapter 15 Number: Gray Code and the Towers of Hanoi -- 15.1 Introduction -- 15.2 Binary Numbers and Gray Code -- 15.3 Gray Code and Rational Numbers -- 15.4 Gray Code and Prime Numbers -- 15.5 Towers of Hanoi -- 15.6 The TOH Sequence Divisibility and Self-replication -- 15.7 Conclusion -- Appendix 15. A -- 15.A1 Converting between Binary and Gray Code -- 15.A2 Converting from Binary to TOH Position -- Chapter 16 Gray Code Sets and Logic -- 16.1 Introduction -- 16.2 Set Theory -- 16.3 Mathematical Logic -- 16.4 Higher Order Venn diagrams -- 16.5 Karnaugh Maps -- 16.6 Karnaugh Maps and n-dimensional Cubes -- 16.7 Karnaugh Maps and DNA -- 16.8 Laws of Form -- 16.9 Conclusion -- Chapter 17 Chaos Theory: A Challenge to Predictability -- 17.1 Introduction -- 17.2 The Logistic Equation -- 17.3 Gray Code and the Dynamics of the Logistic Equation -- 17.4 Symbolic Dynamics -- 17.5 The Morse-Thue Sequence -- 17.6 The Shift Operator -- 17.7 Conclusion -- Appendix 17. A -- Chapter 18 Fractals -- 18.1 Introduction -- 18.2 Historical Perspective -- 18.3 A Geometrical Model of a Coastline -- 18.4 Geometrically Self-Similar Curves.

18.5 Self-Referentiality of Fractals -- 18.6 Fractal Trees -- 18.7 Fractals in Culture -- 18.8 Conclusion -- Chapter 19 Chaos and Fractals -- 19.1 Introduction -- 19.2 Chaos and the Cantor Set -- 19.3 Mandelbrot and Julia Sets -- 19.4 Numbers and Chaos: The Case of c = 0 -- 19.5 Dynamics for Julia Sets with c # 0 -- 19.6 Universality -- 19.7 The Mandelbrot set Revisited -- 19.8 A Mandelbrot Set Crop Circle -- 19.9 Complexity -- 19.10 Conclusion -- Chapter 20 The Golden Mean -- 20.1 Introduction -- 20.2 Fibonacci Numbers and the Golden Mean -- 20.3 Continued Fractions -- 20.4 The Geometry of the Golden Mean -- 20.4.1 The golden rectangle -- 20.4.2 The pentagon -- 20.4.3 Golden triangles -- 20.4.4 Golden diamonds -- 20.4.5 Brunes star -- 20.5 Wythoffs Game -- 20.6 A Fibonacci Number System -- 20.7 Binary and Rabbit "Time Series" -- 20.8 More About the Rabbit Sequence -- 20.9 Conclusion -- Chapter 21 Generalizations of the Golden Mean - I -- 21.1 Introduction -- 21.2 Pascal's Triangle Fibonacci and other n-bonacci Sequences -- 21.3 n-Bonacci Numbers -- 21.4 n-Bonacci Distributions -- 21.5 A General Formula for Limiting Ratios of n-Bonacci Sequences -- 21.6 Conclusion -- Chapter 22 Generalizations of the Golden Mean - II -- 22.1 Introduction -- 22.2 Golden and Silver Means from Pascal's Triangle -- 22.3 Lucas' Version of Pascal's Triangle -- 22.4 Silver Mean Series -- 22.5 Regular Star Polygons -- 22.6 The Relationship between Fibonacci and Lucas Polynomials and Regular Star Polygons -- 22.7 The Relationship between Number and the Geometry of Polygons -- 22.8 Additive Proporties of the Diagonal Lengths -- 22.9 The Heptagonal System -- 22.10 Self-Referential Properties of the Silver Mean Constants -- 22.11 Conclusion -- Appendix 22.A Generalizations of the Vesica Pisces -- Chapter 23 Polygons and Chaos -- 23.1 Introduction.

23.2 Edge Cycles of Star Polygons.