The mathematics of harmony from Euclid to contemporary mathematics and computer science
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, "Ch. 1. The golden section. 1.1. Geometric definition of the golden section. 1.2. Algebraic properties of the golden mean. 1.3. The algebraic equation of the golden mean. 1.4. The golden rectangles and the golden brick. 1.5. Decagon : connection of the golden mean to the number [symbol]. 1.6. The golden right triangle and the golden ellipse. 1.7. The golden isosceles triangles and pentagon. 1.8. The golden section and the mysteries of Egyptian culture. 1.9. The Golden section in Greek culture. 1.10. The golden section in Renaissance art. 1.11. De Divina proportione by Luca Pacioli. 1.12. A proportional scheme of the golden section in architecture. 1.13. The golden section in the art of 19th and 20th centuries. 1.14. A formula of beauty. 1.15. Conclusion -- ch. 2. Fibonacci and Lucas numbers. 2.1. Who was Fibonacci? 2.2. Fibonacci's rabbits. 2.3. Numerology and Fibonacci numbers. 2.4. Variations on Fibonacci theme. 2.5. Lucas numbers. 2.6. Cassini formula. 2.7. Pythagorean triangles and Fibonacci numbers. 2.8. Binet formulas. 2.9. Fibonacci rectangle and Fibonacci spiral. 2.10. Chemistry by Fibonacci. 2.11. Symmetry of nature and the nature of symmetry. 2.12. Omnipresent phyllotaxis. 2.13. \"Fibonacci Resonances\" of the genetic code. 2.14. The golden section and Fibonacci numbers in music and cinema. 2.15. The music of poetry. 2.16. The problem of choice : will Buridan's donkey die? 2.17. Elliott waves. 2.18. The outstanding Fibonacci mathematicians of the 20th century. 2.19. Slavic \"golden\" group. 2.20. Conclusion -- ch. 3. Regular polyhedrons. 3.1. Platonic solids. 3.2. Archimedean solids and star-shaped regular polyhedra. 3.3. A mystery of the Egyptian calendar. 3.4. A Dodecahedron-Icosahedron doctrine. 3.5. Johannes Kepler : from \"Mysterium\" to \"Harmony\". 3.6. A regular icosahedron as the main geometrical object of mathematics. 3.7. Regular polyhedra in nature and science. 3.8. Applications of regular polyhedrons in art. 3.9. Application of the golden mean in contemporary art. 3.10. Conclusion -- ch. 4. Generalizations of Fibonacci numbers and the golden mean. 4.1. A combinatorial approach to the harmony of mathematics. 4.2. Binomial coefficients and Pascal triangle. 4.3. The generalized Fibonacci p-numbers. 4.4. The generalized golden p-sections. 4.5. The generalized principle of the golden section. 4.6. A generalization of Euclid's theorem II.11. 4.7. The roots of the generalized golden algebraic equations. 4.8. The generalized golden algebraic equations of higher degrees. 4.9. The generalized Binet formula for the Fibonacci p-numbers. 4.10. The generalized Lucas p-numbers. 4.11. The \"Metallic Means Family\" by Vera W. de Spinadel. 4.12. Gazale formulas. 4.13. Fibonacci and Lucas m-numbers. 4.14. On the m-extension of the Fibonacci and Lucas p-numbers. 4.15. Structural harmony of systems. 4.16. Conclusion -- ch. 5. Hyperbolic Fibonacci and Lucas functions. 5.1. The simplest elementary functions. 5.2. Hyperbolic functions. 5.3. Hyperbolic Fibonacci and Lucas functions (Stakhov-Tkachenko's definition). 5.4. Integration and differentiation of the hyperbolic Fibonacci and Lucas functions and their main identities. 5.5. Symmetric hyperbolic Fibonacci and Lucas functions (Stakhov-Rozin definition). 5.6. Recursive properties of the symmetric hyperbolic Fibonacci and Lucas functions. 5.7. Hyperbolic properties of the symmetric hyperbolic Fibonacci and Lucas functions and formulas for their differentiation and integration. 5.8. The golden shofar. 5.9. A general theory of the hyperbolic functions. 5.10. A puzzle of phyllotaxis. 5.11. A geometric theory of the hyperbolic functions. 5.12. Bodnar's geometry. 5.13. Conclusion -- ch. 6. Fibonacci and golden matrices. 6.1. Introduction to matrix theory. 6.2. Fibonacci Q-matrix. 6.3. Generalized Fibonacci Q[symbol]-matrices. 6.4. Determinants of the Q[symbol]-matrices and their powers. 6.5. The \"Direct\" and \"Inverse\" Fibonacci matrices. 6.6. Fibonacci G[symbol]-matrices. 6.7. Fibonacci Q[symbol]-matrices. 6.8. Determinants of the Q[symbol]-matrices and their powers. 6.9. The golden Q-matrices. 6.10. The golden G[symbol]-matrices. 6.11. The golden genomatrices by Sergey Petoukhov. 6.12. Conclusion -- ch. 7. Algorithmic measurement theory. 7.1. The role of measurement in the history of science. 7.2. Mathematical measurement theory. 7.3. Evolution of the infinity concept. 7.4. A constructive approach to measurement theory. 7.5. Mathematical model of measurement. 7.6. Classical measurement algorithms. 7.7. The optimal measurement algorithms originating classical positional number systems. 7.8. Optimal measurement algorithms based on the arithmetical square. 7.9. Fibonacci measurement algorithms. 7.10. The main result of algorithmic measurement theory. 7.11. Mathematical theories isomorphic to algorithmic measurement theory. 7.12. Conclusion -- ch. 8. Fibonacci computers. 8.1. A history of computers. 8.2. Basic stages in the history of numeral systems. 8.3. Fibonacci p-codes. 8.4. Minimal form and redundancy of the Fibonacci p-code. 8.5. Fibonacci arithmetic : the classical approach. 8.6. Fibonacci arithmetic : an original approach. 8.7. Fibonacci multiplication and division. 8.8. Hardware realization of the Fibonacci processor. 8.9. Fibonacci processor for noise-tolerant computations. 8.10. The dramatic history of the Fibonacci computer project. 8.11. Conclusion -- ch. 9. Codes of the golden proportion. 9.1. Numeral systems with irrational bases. 9.2. Some mathematical properties of the golden p-proportion codes. 9.3. Conversion of numbers from traditional numeral systems into the golden p-proportion codes. 9.4. Golden arithmetic. 9.5. A new approach to the geometric definition of a number. 9.6. New mathematical properties of natural numbers (Z- and D-properties). 9.7. The F- and L-codes. 9.8. Number-theoretical properties of the golden p-proportion codes. 9.9. The golden resistor dividers. 9.10. Application of the Fibonacci and golden proportion codes to digital-to-analog and analog-to-digital conversion. 9.11. Conclusion -- ch. 10. Ternary mirror-symmetrical arithmetic. 10.1. A ternary computer \"setun\". 10.2. Ternary-symmetrical numeral system. 10.3. Ternary-symmetrical arithmetic. 10.4. Ternary logic. 10.5. Ternary mirror-symmetrical representation. 10.6. The range of number representation and redundancy of the ternary mirror-symmetrical numeral system. 10.7. Mirror-symmetrical summation and subtraction. 10.8. Mirror-symmetrical multiplication and division. 10.9. Typical devices of ternary mirror-symmetrical processors. 10.10. Matrix and pipeline mirror-symmetrical summators. 10.11. Ternary mirror-symmetrical digit-to-analog converter. 10.12. Conclusion -- ch. 11. A new coding theory based on a matrix approach. 11.1. A history of coding theory. 11.2. Non-singular matrices. 11.3. Fibonacci encoding/decoding method based upon matrix multiplication. 11.4. The main checking relations of the Fibonacci encoding/decoding method. 11.5. Error detection and correction. 11.6. Redundancy,
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