By Alexey P. Stakhov, Scott Anthony Olsen
This quantity is as a result the author's 4 many years of analysis within the box of Fibonacci numbers and the Golden part and their purposes. It offers a vast creation to the attention-grabbing and gorgeous topic of the "Mathematics of Harmony," a brand new interdisciplinary path of contemporary technology. This path has its origins in "The parts" of Euclid and has many unforeseen purposes in modern arithmetic (a new method of a historical past of arithmetic, the generalized Fibonacci numbers and the generalized golden proportions, the "golden" algebraic equations, the generalized Binet formulation, Fibonacci and "golden" matrices), theoretical physics (new hyperbolic types of Nature) and desktop technological know-how (algorithmic dimension idea, quantity platforms with irrational radices, Fibonacci pcs, ternary mirror-symmetrical mathematics, a brand new idea of coding and cryptography in response to the Fibonacci and "golden" matrices).The ebook is meant for a large viewers together with arithmetic academics of excessive colleges, scholars of schools and universities and scientists within the box of arithmetic, theoretical physics and computing device technological know-how. The ebook can be utilized as a sophisticated textbook through graduate scholars or even formidable undergraduates in arithmetic and machine technology.
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Extra info for The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science (Series on Knots and Everything 22)
Example text
The linear equation, in general form, can be written as ax + b = 0, where a and b are some numbers and a ≠ 0. This equation has one solution: x = −b / a ; that is, the linear equation has only one root. 25) ax 2 + bx + c = 0, where a ≠ 0. The rules for the solution of algebraic equations of the first and second degree were well known in antiquity. 26) . 25) — which is defined by the following formula: x1,2 = D = b2 − 4ac. 26) gives exactly two real roots. If D = 0, then x = −b / 2a , and we say that Eq.
22) τ = 1+ 1+ τ . If we continue such substitution ad infinitum, we will get another re markable representation of the golden mean τ in “radicals”: τ = 1 + 1 + 1 + 1 + ... ” In this respect, mathematical creative work (aspiration for “aesthetic” expression of mathematical results), is similar to the creative activity of a composer or poet, because their main aim is to find perfect musical or poetic forms that give rise to aesthetic pleasure. 23) give us an aesthetic pleasure, arousing a feeling of rhythm and harmony when we begin to think of the infinite repeatability of the same simple mathematical elements in formulas for τ.
129 from Constructions collection credit Astrid Fitzgerald Homage to Pythagoras credit Marion Drennen This page intentionally left blank Part I Part I. 1. 1. A Problem of the Division in the Extreme and Mean Ratio (DEMR) The Elements of Euclid is one of the best known mathematical works of ancient science. , it contains the main theories of ancient mathematics: elementary geometry, number theory, algebra, the theory of proportions and ratios, methods of calculations of areas and volumes, etc. Euclid, in this work, systematized a 300 year period of de velopment of Greek mathematics, and this work created a strong base for the further development of mathematics.