Abstract

Symmetry: Culture and Science
Volume 33, Number 3, pages 197-199 (2022)
https://doi.org/10.26830/symmetry_2022_3_197

EDITORIAL

Very briefly, and deliberately omitting the historical facts and details due to a lack of place, we have that more than eight centuries have elapsed since Leonardo of Pisa or Fibonacci introduced his famous sequence to Western mathematics and even much more centuries passed since Euclid mentioned the golden ratio for the first time. It appears today that the above sequence and ratio have not aged a bit. Especially these recent years, these two beautiful mathematical objects, and their derivatives and generalizations, have gained a renewed interest in a large number of areas in Science

This Special Issue is a collection of eight papers, all revolving around the lively research subject dealing with the Fibonacci sequence of numbers and the closely related golden ratio. The occurrence of these superb mathematical objects is seen in many recent works and is of considerable importance and interest to a large number of researchers from physics, quantum physics, general relativity, astrophysics, astronomy, chemistry, biology, architecture and so on. A quick search through the Cornell University website arXiv.org, (an online archive of preprint and postprint manuscripts in physics, mathematics,biology, etc.) and using only the two keywords ”Fibonacci” and ”golden ratio” reveals a wealth of works concerning these two mathematical objects. Let us mention, only two examples, from many. First, the golden ratio and the Fibonacci sequence have been shown to be relevant in the study of the phenomenon of quantum entanglement in nonlinear crystals. Second, the golden ratio, and its cousin the “silver” ratio, occur in the algebraic geometry of ???ଶ (Recall that the AdS/CFT correspondence is the most studied area in theoretical physics ever.)

As this Special Issue is open even to dissonant opinions, we have the paper by Huylebrouck who shows that, contrary to the claim of prominent authors, the shape of a Nautilus shell is not described by the well-known circular arcs inscribed in squares with Fibonacci numbers as sides. Instead of the famous golden ratio 1.618..., which is associated with the “most beautiful rectangle”, as many people say, he introduces a new one, 1.355, which he calls “Meta Divine Number”, still connected to, but not the golden ratio, as a new ”myth”.

The paper by Abramovich and Freiman is focused on educational aspects. It shows the potential of technology, physics and digital in experimental mathematics with a particular emphasis on the golden ratio and the connected mathematical extensions as the regular pentagons and decagons. This would greatly help teachers. The content of the paper can be used in various teaching activities going from teacher candidates, and undergraduate mathematics majors as well to school children interested in mathematics and applications to real life.

The paper by Petoukhov, Petukhova and Svirin is devoted to an extension of the usual Fibonacci sequence to the 2-dimensional complex and hyperbolic cases, having in mind other possible multidimensional generalizations. In particular, as applications, they consider phyllotaxis in the light of the defined complex and hyperbolic Fibonacci numbers.

Stepanyan, Savkin and Nechipurenko consider, in their paper, various approaches to DNA sonification. For example, they show how it is possible to simplify the perception of complex genetic information via algorithms of visualisation and sonification of nucleotide sequences.

The paper by Négadi introduces a new numeric form of the genetic code matrix. This matrix, using, in particular, the Fibonacci sequence like some sort of entry key, is shown to encode, as a “Rosetta stone”, the mathematical and chemical structures of the twenty amino acids and more. A link with recent works, emphasizing the prominent role of the three amino acids each coded by six codons, and also with a recent work by him, using selected irreducible representations of the Symmetry group S 23.

Ji, in his interesting paper, proposes to consider two types of golden ratios. A first one, he calls qualitative or phenomenal and widely observed in nature and culture, and a second one, he calls quantitative, precise, mathematical, as it appears in geometry and algebra. He next introduces “causality” and “codality” as two possible relations between these two types and, in the latter case, a category theoretical argument based on bio-evolutionary data is advanced to support it. Arguments from semiotics are also introduced to describe the above relations between the two types considered as “signs”.

The paper by Voinova considers the Fibonacci sequence in acoustics. She presents a brief review of several innovative solutions and engineered quasiperiodic systems based on Fibonacci sequences with various applications in acoustics (metamaterials, Fibonacci multilayers and superlattices, phononic crystals, quasiperiodic heterostructures and quasicrystals).

The paper by Kabai explores the occurrence of the golden ratio in geometry, using Wolfram Mathematica images. From the hundreds in his personal collection, he presents seventeen ones, all, but not the only ones, connected to the golden ratio.

Tidjani Négadi
Department of Physics
Faculty of Exact and Applied Science
University Oran1, Oran, Algeria
Guest Editor