Fibonacci Numbers and Golden Section in the Higher School
But if we started our reform in the secondary school, where we attempted to enter the elementary information on the Fibonacci numbers, the golden section and their applications in Nature and Art, this reform should be continued in the higher educational institutions.
The first proposal is to enter a small course "Harmony of systems" for the students of colleges and universities of all specialties. And ours "Virtual Museum of Harmony and Golden Section" also is some Virtual Lab and Visual Aids for such course.
The program of the course "Harmony of systems"
Subject 1. Introduction into the "Golden Section".
What is the golden section? Geometrical definition. "Golden proportion" equation. Mathematical properties of the golden proportion Geometrical properties of the golden section. How to construct the golden section with the help of a rule and divider? The golden section and the "sacred" geometrical figures. Connection with a circumference. "Golden" rectangle. Decagon and pentagon (pentagram). Law of the "golden" cup. "Golden" triangles. "Golden" spirals. "Platonic Solids" and their connection to the golden section. Numeric characteristics of the dodecahedron and icosahedron.
Subject 2. Fibonacci and Lucas numbers.
Problem of "rabbits reproduction". Fibonacci'd recurrent formula. Lucas numbers. Mathematical properties of Fibonacci and Lucas numbers. Connection to the golden section. Binet's formulas.
Subject 3. Golden Section in the Egyptian culture.
History of the Egyptian pyramids. Proportions of the Egyptian pyramids. Golden section in Cheops pyramid. "Sacred" triangle in Chefre pyramid. Dodecahedral-icosahedral idea. Implementation of the cyclical processes of the Solar system (12-year's cycle of Jupiter, 30-year's cycle of Saturn, main 60-year's cycle of the Solar system) in the dodecahedron and icosahedron. New hypothesis of the Egyptian calendar origin. Why we have 12 Zodiac Signs? Why the Babylonian notation had the basis 60? Why we divide a circle on 360°?
Subject 4. Golden Section in the Ancient Greek culture.
Sources of the Ancient Greek culture and its connection to the Egyptian culture. Pytsagor and his role in knowledge transmission of from the Egyptian culture to the Greek culture. Pythagorean theorem. Pythagorean number theory. Pythagorean harmony theory. Pythagorean doctrine about numeric harmony of the Universe. A role of the golden section in the Greek architecture and sculpture. "Parthenon": its history and proportions.
Subject 5. Golden Section in the epoch of the Middle Ages and Renaissance.
Fibonacci and his role in a history of mathematics as the mathematical knowledge messenger from the Arabian culture to the West-European mathematics. "Liber abaci" by Fibonacci as the mathematical encyclopedia of the Middle Ages epoch. Revival of interest in the Greek culture in the Italian Renaissance epoch. Aesthetic views and researches of Leonardo da Vince, Alberti, Durer.
Usage of the "golden" triangle for composite construction of the famous "Joconda". "Golden" spirals in Rafael's pictures. Mathematical works by Luca Pachioli. Pachioli's book "De Divina Proportione". Kepler's model of the Solar system based on the "Platonic solids".
Subject 6. Revival of interest in the golden section and Fibonacci numbers in the 19th century.
Mathematical researches of the French mathematicians Binet and Lucas. Lucas numbers and Binet's formulas. "The Overall law of proportionality" by the German scientist Zeizing. Fechner's psychological experiments. Great geometer Felix Klein about the role of the "Platonic Solids" in development of mathematics.
Subject 7. Golden Section in morphology of plants, animals and a man.
Symmetry laws. The principle of the least action in the nature. Manifestation of the golden section in the animal world. Shells used the "golden" equiangular spiral. Sea asters and urchins with the pentagonal symmetry.
Manifestation of the golden section in the plant world. Fibonacci numbers and the botanic "law of phyllotaxis".
Golden section and a Man. Proportions of man's body. Dorifor's statue. Human figures from famous Leonardo da Vince's notebooks. Man's and woman's proportions by Zeising. Proportions of tri-limbed links of man's body. Brain's rhythms. Fibonacci numbers in the mammals cardiac activity.
Subject 8. Golden Section in architecture and fine art.
Geometrical harmony in the architecture. A measure in the antique and ancient architecture. The stiffened music of the Russian temples. Harmonic analysis of the Smolny cathedral in St.-Petersburg, Great Pechora church in Kiev. Usage of the golden rectangle in painting. Harmonic analysis of the Russian artists pictures (Repin, Shichkin, Surikov, Ivanov, Ge, Rokotov, Vasil'ev).
Subject 9. Golden Section in music, poetry, cinema.
Fundamentals of the musical forms organization. Sabaneev's researches. "Shopen's etudes in the lighting the golden section". Rosenov's researches. "Golden section" in Beethoven' "Appassionato" and Tchaikovsky's music.
Golden section in poetry. Lwrmontov's poetry. Harmonic analysis of poem "Borodino", "Aircraft", "Demon", "Three palms ". Pushkin's poetry. Harmonic analysis of poems "Ruslan and Ludmila", "Poltava". The sizes of Pushkin's poems. Harmonic analysis of Shota Rustavely's poem "King in tiger skin". Golden section in the cinema. Eisenstein's film "Potemkin's Armadillo".
Subject 10. The Solar system. The form and surface of the Earth.
The resonant theory of the Solar system and golden section in the planet's cycles. Natural numbers of Cosdmos. Solar activity.
The form of the Earth and constitution of its surface. A hypothesis of the Russian geologist Kislitsin about the dodecahedral-icosahedral form of the Earth.
Subject 11. Chemistry "on Fibonacci".
The law of multiple proportions and stabilityof chemical compounds. Oxides of uranium and chromium based on Fibonacci numbers. Searching the golden section in Mendeleev's table.
Subject 12. Fibonacci numbers and golden section in modern mathematics.
Binomial coefficients and Pascal triangle. Fibonacci numbers in Pascal triangle. The generalized Fibonacci numbers and generalized golden sections. Concept of Fibonacci matrices. Concept of the hyperbolic Fibonacci and Lucas functions.
Subject 13. Computers based on Fibonacci numbers and golden section.
Fibonacci codes. Fibonacci arithmetic. Fibonacci computers. Codes of the golden proportion. "Golden" arithmetic. Coding theory based on Fibonacci matrices.
Subject 14. Modern scientific discoveries based on the golden section.
Shechtman's quasi-crystals. Bodnar's theory of phyllotaxis. Soroko's law of structural system harmony. "Elliott's Wave Principle and a new science of socionimics.