We give in the article a survey of the new results in the «Mathematics of the Golden Section», going back in its origin to Euclid’s Elements, namely, a new class of hyperbolic functions, a generalization of Euclid’s Theorem II,11, a new class of hyperbolic functions — hyperbolic Fibonacci and Lucas functions, the generalized Fibonacci numbers, the generalized golden proportions, the generalized principle of the golden section, the golden algebraic equations, the generalized Binet formulas, the generalized Lucas numbers, Fibonacci matrices, the «golden» matrices. We consider applications of these mathematical results in number theory, measurement theory, computer arithmetic, coding theory and cryptography.

In the beginning of twentieth century classical natural philosophy ( classical physics ) was destroyed and replaced by so called non classic mathematical physics consisting of theor y of relativ ity and quantum mechanics. New non classic physics is based on postulates and faith, so it is more religion , than a scienc e. This article is a fragment of a new book in which development of natural ideas classical physics is presented.

We consider a new class of square matrices called the «golden» matrices. They are a generalization of the classical Fibonacci Q-matrix for continuous domain. The «golden» matrices can be used for creation of a new kind of cryptography called the «golden» cryptography. The method is very fast and simple for technical realization and can be used for the cryptographic protection of different telecommunication systems (including Internet) functioning in real scale of time.

SEVENTY-FIVE years ago this month, The New York Times reported that Albert Einstein had completed his unified field theory — a theory that promised to stitch all of nature's forces into a single, tightly woven mathematical tapestry. But as had happened before and would happen again, closer scrutiny revealed flaws that sent Einstein back to the drawing board. Nevertheless, Einstein's belief that he'd one day complete the unified theory rarely faltered. Even on his deathbed he scribbled equations in the desperate but fading hope that the theory would finally materialize. It didn't.

Based on the PhD thesis of Anne Bulckens, the proportional system of the Parthenon is examined. With the appropriate choice of a module and the length of a «Parthenon foot», all dimensions are shown to be integers. These integer values are shown to be related to the musical scale of Pythagoras. Among the many musical relationships expressed by this structure, this paper focuses on a pentatonic scale made up of lengths, widths and heights of the outer temple and the inner temple or cella.

The logic by which the creator of the Nefertiti statuette was guided is clear and simple. Harmony is the prerogative of the divine order which dominates the Universe, and geometry is a means of its expression. The Queen is given to people by God. Hence, her image, personifyng the world’s wise tranquillity must be full of geometrical perfection to bear the stamp of high and irreproachable harmony, beauty and clearness. Such is, in fact, the whole philosophy of ancient Egyptian art, which glorified the eternal, the measured, the perfect in the ever changing being.

Before starting, I offer the following overview, the details of which will be discussed in the rest of the article. An application of abductive reasoning to Plato's puzzles in the dialogues leads to the solution that the Divided Line in the Republic is constructed using a series of Golden Cuts (i.e., divisions in extreme and mean ratio). This leads to the discovery that there is a more primitive form than the √2 and √3 ratios (the roots inherent in the elementary triangles of the Timaeus), and that this form is based in the Golden Section.

The main goal of the present article is to consider the harmony mathematics from the point of view of the sacral geometry and to show how it can be used in this field. We also consider some secrets of the Egyptian civilization that have relation to the golden section and platonic solids. Briefly, this is considered to be the main concepts involved in harmony mathematics and its application to the sacral geometry.

We consider two important generalizations of the golden proportion: golden p-proportions [Stakhov A.P.] and «metallic means» [Spinadel V.W.]. We develop a constructive approach to the theory of real numbers that is based on the number systems with irrational radices (Bergman's number system and Stakhov's codes of the golden p-proportions). It follows from this approach ternary mirror-symmetrical arithmetic that is the basis of new computer projects.

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into «continuous» theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers.

The goal of the present article is to develop the «continues» approach to the recurrent Fibonacci sequence. The main result of the article is new mathematical model of a curve-linear space based on a special second-degree function named «The Golden Shofar».

The special case of the (p+1)th degree algebraic equations of the kind x^p+1 = x^p + 1 (p = 1,2,3,...) is researched in the present article. For the case p = 1, the given equation is reduced to the well-known Golden Proportion equation x² = x + 1. These equations are called the golden algebraic equations because the golden p-proportions τ_p, special irrational numbers that follow from Pascal's triangle, are their roots. A research on the general properties of the roots of the golden algebraic equations is carried out in this article.

We consider a new class of square Fibonacci (p + 1) × (p + 1)-matrices, which are based on the Fibonacci p-numbers (p = 0,1,2,3,...), with a determinant equal to +1 or — 1. This unique property leads to a generalization of the «Cassini formula» for Fibonacci numbers. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. In contrast to classical redundant codes a basic peculiarity of the method is that it allows to correct matrix elements that can be theoretically unlimited integers. For the simplest case the correct ability of the method is equal 93.33% which exceeds essentially all well-known correcting codes.

The new continuous functions for the Fibonacci and Lucas p-numbers using Binet formulas are introduced. The article is of a fundamental interest for Fibonacci numbers theory and theoretical physics.

There have been obtained experimental results on remote effects of rotating objects on readings of semiconductor gamma – ray spectrometric equipment and devices. It is shown that the trace shift of statistic distribution is the function of velocity changes in semiconductor detector’s charges accumulation. In its turn, the accumulation rate is subject to a physical field of non-electromagnetic origin created in the result of rotation.

Modern natural science requires the development of new mathematical apparatus. The generalized Fibonacci numbers or Fibonacci p-numbers (p = 0,1,2,3,...), which appear in the "diagonal sums" of Pascal's triangle and are assigned in the recurrent form, are a new mathematical discovery. The purpose of the present article is to derive analytical formulas for the Fibonacci p-numbers. We show that these formulas are similar to the Binet formulas for the classical Fibonacci numbers. Moreover, in this article, there is derived one more class of the recurrent sequences, which is defined to be a generalization of the Lucas numbers (Lucas p-numbers).

The «Dichotomy Principle» and the classical «Golden Section Principle» are two of the most important principles of Nature, Science and also Art. The Generalized Principle of the Golden Section that follows from studying the diagonal sums of the Pascal triangle is a sweeping generalization of these important principles. This underlies the foundation of «Harmony Mathematics», a new proposed mathematical direction. Harmony Mathematics includes a number of new mathematical theories: an algorithmic measurement theory, a new number theory, a new theory of hyperbolic functions based on Fibonacci and Lucas numbers, and a theory of the Fibonacci and «Golden» matrices.

As is well known, mathematics is one of the outstanding creations of the human intellect; a result of centuries of intensive and continuous creative work of man's geniuses. What is the goal of mathematics? The answer is not simple. Probably, the goal of mathematics is to discover «mathematical laws of the Universe» and to construct models of the physical world. It is clear that the progress of the human society depends on the knowledge of these laws.

A graph, G, is a mathematical object composed of vertices, j V}, and edges {E}, where an edge spans a pair of vertices (Harary, 1969). A matching of a graph is a set of edges of G such that no two of them share a vertex in common. If a graph with even n = |V| has a matching with л/2 edges it is called a perfect matching graph. The number of possible perfect matchings of G is the perfect matching number, or Kekule number, K(G). Although a tree graph has at most one Kekule structure, a number of interesting features have been found for the K{G) numbers of polycyclic graphs (HOSOYA, 1986).

It has already been shown by Lucas (1876) that the Fibonacci numbers can be obtained from the Pascal's triangle by rotation and addition in a certain way. The golden ratio can then be asymptotically obtained by taking the ratio of consecutive Fibonacci numbers, whose graph-theoretical aspects have been pointed out by the present author (Hosoya, 1971, 1973) in terms of the topological index, Z, which is the sum of the non-adjacent numbers for a given graph. Similar properties of the Lucas numbers related to the golden ratio have also been demonstrated.

We now have a situation in genetics, molecular biology, and medicine in general, that is simultaneously both paradoxical and promising. Long ago, science decided to investigate the genetic codes of human beings.

Science has now completed the 10 years long effort of mapping the DNA sequences of human beings, called the Genome. All of the letters and sequences of the DNA codes of human beings are now known.

I have followed the recent correspondence of several authors about your work, and I want to express my own opinion also. From these correspondents, which sometimes are scientific artiles by themselves, I can only suggest that your work, both in experiment and in theory, is of high interest and importance. I can only suggest that your work be kept going on in spite of the difficulties you are encountering and I personally have a high opinion on it.

Dr. Gennady Shipov is one of the world’s leading physicists in Torsion-Physics research. He joins us to talk about the fundamentals of this emerging branch of scientific discovery, and provides some unique insight into how we can turn the fundamental forces of nature towards the goal of advanced propulsion…

Gravity is nature's most mysterious force — or is it? Einstein's Relativity suggested that it's not a force at all, but instead a curvature of time & space. His later research into Unified Field Theory physics extended this notion with the concept of torsion, which many physicists believe has the power to «uncurve» space and make possible a new generation of advanced propulsion devices.

Consider Cn spaces where the exterior differentiation of a vector basis of C2 functions leads to a Cartan matrix of exterior differential 1-forms. A second exterior differentiation leads to a Cartan matrix of curvature 2-forms, which must evaluate to zero, by the Poincare lemma. The Cartan connection matrix can be decomposed in to two parts, one part based on a metric (Christoffel) connection, and the other part on a residue matrix of 1-forms such that [С] = [Γ] + [T]. The exterior differential of the composite connection must vanish such that the curvature 2-forms produced by the metric component must be balanced by all other matrices of 2-forms coming from the Residue connection and interaction 2-forms generated by exterior products of the [Γ] and [Γ].

Abstract: In this paper, we present the Metallic Means Family (MMF), being the most paramount of its members, the Golden Mean f and in the second place, the Silver Mean s _Ag. We call them a family because, aside from carrying the name of a metal – the Golden Mean, the Silver Mean, the Copper Mean, the Bronze Mean, the Nickel Mean – they enjoy common mathematical properties that confer them a fundamental importance in modern investigations. Among these applications, we have chosen the analysis of the forbidden symmetries in a quasicrystal.

Dr. Vera W. de Spinadel has got her PhD in Mathematics at the University of Buenos Aires, Argentina, in 1958. At present, she is Full Consultant Professor of Mathematics at the University of Buenos Aires. This is her second period since she got this degree in 1995 until 2002 and it has been renewed until 2009. Since 1995, she is the Director of the research Center on Mathematics & Design MAyDI, having received many I&D grants and personal grants to develop the activitiesof many research groups. In April 2005, she has officially inaugurated the Mathematics & Design Laboratory at the Faculty of Architecture, Design and Urban Planning, University of Buenos.

We invite philosophers, mathematicians, experts of all areas of science and education and also students of universities to take active part in the offered international action of scientific cooperation that is directed on the formation of harmonious mutual relations "Person-Society-Nature". The primary publication of materials concerning to the "Harmony Mathematics" and their discussion are realized on the site www.trinitas.ru and can be duplicated on any other sites and in different languages.

Einstein believed that one of the main problems in unified field theory is the one of the geometrization of the energy-momentum tensor of matter on the right-hand side of his equations. This problem can be solved using the geometry of absolute parallelism and Cartan's structural equations in this geometry:

For 317 years we have been applying Newton's mechanics to explain non-relativistic mechanical experiments on the "bench table". Although Newton's mechanics has been generalized three times: the special relativity theory, general relativity theory, and quantum mechanics, there remains a possibility for its further generalization.

Dr. Shipov has generalized the ordinary four-dimensional Relativity Theory. He showed that the right-hand sides of the Einstein field equations for gravity and the equations of general-relativistic electrodynamics can be geometrized successfully, if one uses not a Riemannian geometry but the geometry of absolute parallelism. The new field equations he suggests were written as

Theoretical bases of a quality monitoring. Vibroacoustik the information from own vibrations bells products on the character carries relaccions character. At the description of this information in time it is possible to allocate the following periods of life Vibroacoustik information:

In response to your call for the ideas and information to promote the World Year of Physics, I wish to bring to your attention the revolutionary work of Dr. Gennady Shipov, Academician of the Russian Academy of Natural Science and Director of Science Center of Vacuum Physics.

Since Russian Academy of Science's (RAS) «Commission on fight with Fraud in Science, Pseudoscientists and Torsion Fields» has launched a campaign against the work of Dr. G. Shipov, I would like to respond to your references in websites as well as to ask you several questions: