Beyond measure a guided tour through nature myth and number 1st Edition by Jay Kappraff ISBN 9810247028 978-9810247027

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ISBN 10: 9810247028
ISBN 13: 978-9810247027
Author: Jay Kappraff

This book consists of essays that stand on their own but are also loosely connected. Part I documents how numbers and geometry arise in several cultural contexts and in nature: the ancient musical scale, proportion in architecture, ancient geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of biological forms. The focus is on how certain numbers, such as the golden and silver means, present themselves within these systems. Part II shows how many of the same numbers and number sequences are related to the modern mathematical study of numbers, dynamical systems, chaos, and fractals.

Beyond measure a guided tour through nature myth and number 1st Table of contents:

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 Othe

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 Proble

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

23.3 The Relationship between Polygons and Chaos for the Cyclotomic 7-gon

23.4 Polygons and Chaos for the 7-cyclotomic Polygon

23.5 Polygons and Chaos for Generalized Logistic Equations

23.6 New Mandelbrot and Julia Sets

23.7 Chaos and Number

23.8 Conclusion

Appendix 23.A

Appendix 23.B

Appendix 23.C

Chapter 24 Growth of Plants: A Study in Number

24.1 Introduction

24.2 Three Models of Plant Growth

24.2.1 Coxeter’s model

24.2.2 Van Iterson’s model

24.2.3 Rivier’s model

24.3 Optimal Spacing

24.4 The Gears of Life

24.5 Conclusion

Appendix 24.A The Golden Mean and Optimal Spacing

Chapter 25 Dynamical Systems

25.1 Introduction

25.2 Quasicrystals

25.3 The Ising Problem

25.3.1 The Ising model

25.3.2 The Devil’s Staircase for Ising spins

25.3.3 Spatial Distribution of Spins

25.4 The Circle Map and Chaos

25.5 Mode Locking

25.6 Mode Locking and Natural Resonances

25.7 Mode Locking and the Harmonics of the Musical Scale

25.8 Mode Locking and the Circle Map

25.9 Blackmore’s Strain Energy: A Unifying Concept

25.10 Conclusion

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