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Beyond Measure : a Guided Tour Through Nature, Myth and Number.

Author: Jay Kappraff
Publisher: Singapore : World Scientific Publishing Company, 2002.
Series: K & E series on knots and everything.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Jay Kappraff
ISBN: 9789812778444 9812778446
OCLC Number: 879023686
Description: 1 online resource (616 pages).
Contents: Introduction --
Acknowledgements --
Permission --
pt. I. Essays in geometry and number as they arise in nature, music, architecture and design. ch. 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. ch. 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 2A. Homogeneous coordinates. ch. 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 3A. 3A1. Logarithms and the logarithmic spiral. 3A2. Properties of logarithms. 3A3. Logarithms and the musical scale. Appendix 3B. The Pythagorean comma. Appendix 3C. Vectors. ch. 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. 4.5. Conclusion. Appendix 4A. ch. 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 5A Kepler's ratios. ch. 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 6A. 6A1. Steps to creating a triangular grid of circles. 6A2. Steps to creating a square circle grid. Ch. 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. 12.7. Generation of the flame-hand letters. 12.8. Some commentary on Tenen's proposal. 12.9. Conclusion --
pt. II. Concepts described in part I reappear in the context of fractals, chaos, plant growth and other dynamical systems. ch. 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 13A. ch. 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 14A. Euler's [symbol]-function. Appendix 14B. The relation between continued fraction indices and the little end of the stick problem. Appendix 14C. "Kissing" gears. ch. 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 15A. 15A1. Converting between binary and gray code. 15A2. Converting from binary to TOH position. ch. 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. ch. 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 17A. Ch. 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. ch. 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[symbol]0. 19.6. Universality. 19.7. The Mandelbrot set revisited. 19.8. A Mandelbrot set crop circle. 19.9. Complexity. 19.10. Conclusion. ch. 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. 20.5. Wythoffs game. 20.6. A Fibonacci number system. 20.7. Binary and rabbit "time series". 20.8. More about the rabbit sequence. Conclusion ch. 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. ch. 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 properties of the diagonal lengths. 22.9. The Heptagonal system. 22.10. Self-referential properties of the silver mean constants. 22.11. Conclusion. Appendix 22A. Generalizations of the Vesica pisces. ch. 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 23A. Appendix 23B. Appendix 23C. ch. 24. Growth of plants: a study in number. 24.1. Introduction. 24.2. Three models of plant growth. 24.3 Optimal spacing. 24.4. The gears of life. 24.5. Conclusion. Appendix 24A. The golden mean and optimal spacing. ch. 25. Dynamical systems. 25.1. Introduction. 25.2. Quasicrystals. 25.3. The Ising problem. 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 --
Epilogue --
Bibliography --
Index.
Series Title: K & E series on knots and everything.

Abstract:

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 m.

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