Matila Ghyka – between art and science

Matila Ghyka - naval officer, engineer, mathematician, diplomat, but known as the author of studies on the golden number and rhythm, as well as his memoirs published in French and English. Today he is considered an old-fashioned scholar, with availability for access to ancient culture, from Pythagoras to the scientific knowledge of his life, from the theory of relativity and knowledge of the atom, to Buddhist and Taoist theories, all integrated. in the humanist effort for a better, more just world, without wars and discrimination.

Application of Banach contraction principle to approximate the golden number

This work is devoted to the application of selected fixed point theorems in the problems of convergence of certain sequences to the golden number. It contains the theorem about the fixed point of so-called ψ-contraction specified on the closed interval <a, b> and the local version of Banach Contraction Principle as a conclusion. It will also be used to approximate the golden number.

The Golden Number

The golden number or divine proportion was defined by Euclid. It is sometimes claimed that it was used in classical architecture, but it is not mentioned by Vitruvius, so this seems unlikely. The illustrations for Luca Pacioli’s book The Divine Proportion were drawn by Leonardo. The golden number is related to the structure of polyhedra with five-fold symmetry. Chapter 13 considers some of the properties of the regular and semi-regular or Archimedean polyhedra, and also considers the suggestion that the pupil in the famous painting of Luca Pacioli is a young Albrecht Dürer.

Geometrical analysis of architectural drawnings in the Shah-mosque Isfahan

The golden number has been one of the most important measures of beauty from ancient Greece till now. Although the reason still unclear, it is certain that the golden ratio has a direct correlation with beauty. This nice feeling can come from listening to a musical piece or watching a visual art, which is based on this wondrous number. Although the golden number is more suitable from beauty or harmony’s perspective, for some researches it is a requirement to develop Architectural designs. Therefore, in this article a masterpiece (Shah-mosque Isfahan) was investigated to find its golden relations. This mosque was chosen because of the precise and harmonious proportions. In this research: plan, sections, facade and decorations of ceiling tiles in main dome, were analyzed by Phi-matrix software in which, existence of the golden ratio has been proven undoubtedly.

On the golden number and Fibonacci type sequences

The paper presents, among others, the golden number $\varphi$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein's theorem. To demonstrate the equality , where $f_n$ is $n$-th Fibonacci number also the formula from Corollary \ref{cor1} has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

Multidimensional generalization of phyllotaxis

The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance of the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1]. We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.

A Non-Regular Icosahedron Geometry Satellite Form, Mirror Invested Polyhedro Heliotrope, For Optical Tracking

Working on relationships of three circles in common ratio [4/π or square root of the golden number ] and drawing lines of related tangents, squares and triangles, viewed on the paper plan, a figure having the shape of a section [Hexagonal] similar to that of an Icosahedron or Dodecahedron. This gave me the idea of searching for an existing probable Polyhedron built upon this traced shape. In fact this Polyhedron was built[ 4x scale], whose geometry relates to the Icosahedron and the Dodecahedron. It is a non regular Icosahedron having 12 Isosceli triangles and 8 Equilateral triangles. Mirror triangles cut to size, invested the structure for the configuration of a “Polyhedroheliotrope”Satellite Optical Tracking application.