Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science

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Alexey Stakhov. Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science

Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science

Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science

Contents

Preface to the Three-Volume Book. Continuity in the Development of Science

Pythagoreanism and Pythagorean MATHEM’s

Proclus Hypothesis

The Statements by Alexey Losev and Johannes Kepler

Ancient Greeks Mathematical Doctrine of Nature

Introduction of the Term Mathematics of Harmony

Fibonacci Numbers

Sources of the Present Three-Volume Book

Structure and the Main Goal of the Three-Volume Book

Introduction

About the Author

Stakhov’s Prizes and Awards

Acknowledgments

Scientific cooperation of Alexey Stakhov and Samuil Aranson

Stakhov and Aranson’s Mathematical Monographs in English

Stakhov and Aranson’s Scientific Papers in English

Chapter 1. The Golden Section: History and Applications. 1.1. The Idea of the Universal Harmony in Ancient Greek Science. 1.1.1. What is Harmony?

1.1.2. Numerical harmony of Pythagoreans

1.1.3. The contribution of Heraclitus to the development of the doctrine of Harmony

1.1.4. The musical harmony of Pythagoras and the music of the spheres

1.1.5. Once again about the term of the Mathematics of Harmony

1.2. The Golden Section in Euclid’s Elements

1.2.1. Proposition II.11 of Euclid’s Elements

1.2.2. The second form of the task of the division of segment in the extreme and mean ratio

1.2.3. Comments of Mordukhai-Boltovsky, concerning the golden section

1.2.4. The origin of the term of the golden section

1.3. Proclus Hypothesis and New View on Classic Mathematics and Mathematics of Harmony. 1.3.1. For what purpose did Euclid write his Elements?

1.3.2. The significance of the Proclus hypothesis for the development of mathematics

1.3.3. The Proclus hypothesis and “key” problems of ancient mathematics

1.3.4. How did Euclid use the golden section in his Elements?

1.4. Some Simplest Mathematical Properties of the Golden Ratio. 1.4.1. The simplest algebraic identities for the golden ratio

1.4.2. “Golden” geometric progression

1.4.3. Representation of the golden proportion in the “radicals”

1.5. The Golden Ratio and Chain Fractions. 1.5.1. General information about chain fractions

1.5.2. Representation of the “golden proportion” in the form of a chain fraction

1.6. Equations of the Golden Proportion of the Nth Degree

1.7. Geometric Figures Associated with the Golden Section. 1.7.1. The Golden rectangle

1.7.2. A “double” square

1.7.3. The “golden” rectangular triangle (Kepler’s triangle)

1.7.4. The “golden” ellipse

1.7.5. The golden bowl and the golden isosceles triangle

1.7.6. Decagon: the connection of the “golden section” with the number π

1.8. The Golden Section in Nature. 1.8.1. Pentagonal symmetry in Nature

1.8.2. Pentacle of Venus

1.9. The Golden Section in Cheops Pyramid

1.10. The Golden Section in Ancient Greek Culture. 1.10.1. The magnificent Parthenon

1.10.2. The golden section in Greek sculpture

1.11. Golden Section in the Art of the Renaissance. 1.11.1. The idea of the “divine harmony” in the Renaissance

1.11.2. “David” Michelangelo

1.11.3. The painting “Mona Lisa” by Leonardo da Vinci

Chapter 2. Fibonacci and Lucas Numbers. 2.1. A History of the Fibonacci Numbers. 2.1.1. Who is Fibonacci?

2.1.2. The task of rabbits’ reproduction

2.1.3. Variations on Fibonacci theme

2.2. The Sums of the Consecutive Fibonacci Numbers

2.3. Cassini’s Formula. 2.3.1. Who is Giovanni Cassini?

2.3.2. Cassini’s contribution to the development of the theory of Fibonacci numbers (Cassini’s formula)

2.4. Lucas Numbers. 2.4.1. Who is Lucas?

2.4.2. Generalized Fibonacci numbers

2.4.3. The extended Fibonacci and Lucas numbers

2.5. Binet’s Formulas. 2.5.1. Who is Binet?

2.5.2. Deducing Binet’s formulas

2.5.3. Proof of Cassini formula

2.6. Steinhaus’s “Iron Table”

2.7. Pythagorean Triangles and Their Presentation Through Fibonacci and Lucas Numbers. 2.7.1. Pythagoras theorem

2.7.2. Pythagorean triangles

2.7.3. Fibonacci’s “Pythagorean triangles”

2.7.4. Lucas’ “Pythagorean triangles”

2.8. Fibonacci Numbers in Nature. 2.8.1. Fibonacci rectangles

2.8.2. Fibonacci spirals

2.8.3. Phenomenon of phyllotaxis and Fibonacci numbers

2.9. Fibonacci Numbers and Solution of Hilbert 10th Problem

2.10. Turing and Fibonacci Numbers

2.11. Role of the Fibonacci Numbers Theory in Modern Mathematics. 2.11.1. Fibonacci association

2.11.2. The role of Nikolay Vorobyov in the development of Fibonacci number theory

Chapter 3. Pascal Triangle, Fibonacci p-Numbers and Golden p-Proportions. 3.1. Binomial Theorem

3.2. Pascal Triangle

3.3. Diagonal Sums of Pascal’s Triangle and Fibonacci p-Numbers. 3.3.1. Mathematical discovery of George Polya

3.3.2. Pascal’s rightangular triangle

3.3.3. Pascal’s 1-triangle

3.3.4. Pascal’s p-triangles and Fibonacci p-numbers

3.4. The Extended Fibonacci p-Numbers

3.5. Generalization of the Golden Section Problem. 3.5.1. The limit of the ratio of the neighboring Fibonacci p-numbers

3.5.2. Geometric definition

3.5.3. Some algebraic properties of the golden p-proportion

3.6. Algebraic Equations for the Golden p-Proportion and Vieta’s Formulas. 3.6.1. General properties of the roots of the golden p-proportion equations

3.6.2. Vieta’s formulas for quadratic equations

3.6.3. Vieta’s formulas for the general case

3.6.4. Vieta’s formulas for the roots of the golden section equation

3.6.5. Vieta’s formulas for the equation of the golden p-proportion

3.7. Binet’s Formulas for the Fibonacci p-Numbers. 3.7.1. General approach

3.8. Binet’s Formulas for the Lucas p-Numbers

3.8.1. Binet’s formula for the Fibonacci 2-numbers

3.8.2. Binet’s formula for the Lucas 2-numbers

Chapter 4. Platonic Solids: From Plato’s Cosmology to Fullerenes and Quasicrystals. 4.1. The Golden Section in Platonic Solids

4.1.1. Platonic solids

4.1.2. Numerical parameters of the Platonic solids

4.1.3. The golden section in dodecahedron and icosahedron

4.2. Archimedean Truncated Icosahedron and Stellate Polyhedra. 4.2.1. Archimedean solids

4.2.2. Archimedian truncated icosahedron

4.2.3. Stellate polyhedral

4.3. The Mystery of the Egyptian Calendar. 4.3.1. What is a calendar?

4.3.2. The structure of the Egyptian calendar

4.3.3. Connection of the Egyptian calendar with the numerical characteristics of the dodecahedron

4.3.4. About the Mayan calendar

4.4. Dodecahedral–Icosahedral Doctrine. 4.4.1. The sources of the doctrine

4.4.2. Earth shape

4.5. Johannes Kepler: From “Mystery” to “Harmony” 4.5.1. Misterium Cosmographicum

4.6. Icosahedron as the Main Geometric Object of Mathematics. 4.6.1. Felix Klein

4.6.2. Symmetry groups of regular polyhedra

4.6.3. The role of the icosahedron in the development of modern science

4.7. Usage of the Regular Polyhedra in the Fine Art

4.7.1. Leonardo da Vinci image techniques for regular polyhedra

4.7.2. Polyhedra by Luca Pacioli

4.7.3. Albrecht Dürer

4.7.4. Piero della Francesca

4.7.5. Intarsia art

4.7.6. The sacrament of the last supper by Salvador Dali

4.7.7. Creativity of Maurits Escher

4.7.8. Matjuska Teja Krasek’s World

4.8. Dan Shechtman’s Quasicrystals. 4.8.1. Brief information about Dan Shechtman

4.8.2. What are quasicrystals?

4.8.3. Experimental study of quasicrystals

4.8.4. Penrose’s tiles

4.8.5. Methodological significance of the discovery of quasicrystals

4.9. Fullerenes. 4.9.1. What are “fullerenes”?

4.9.2. A history of discovery

4.9.3. Fullerene applications

4.10. Platonic Solids and New Ideas in the Theory of Elementary Particles. 4.10.1. Icosahedron and elementary particles

4.10.2. Quark icosahedron of Yuri Vladimirov

Conclusions

Bibliography. The books in the Field of the Golden Section, the Fibonacci numbers and the Mathematics of Harmony

The Articles in the Field of the Golden Section, the Fibonacci Numbers and the Mathematics of Harmony

Other Publications

Отрывок из книги

Volume 1

The Golden Section, Fibonacci Numbers, Pascal Triangle, and Platonic Solids

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4.2 Archimedean Truncated Icosahedron and Stellate Polyhedra

4.3 The Mystery of the Egyptian Calendar

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