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Fibonacci and Lucas Numbers and the Golden Ratio in Physics...
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Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: 30 November 2024 | Viewed by 27776

Special Issue Editors

Dr. Tidjani Negadi

E-Mail Website1 Website2
Guest Editor
Physics Department, Faculty of Exact and Applied Sciences, University of Oran1, Oran, Algeria
Interests: theoretical physics; mathematical biology
Dr. Michel Planat

E-Mail Website
Guest Editor
CNRS, Institut FEMTO-ST, Université de Franche-Comté, F-25044 Besançon, France
Interests: topological quantum computing; epigenetics and epitranscriptomics; signal processing; geometry; quantum mechanics; discrete mathematics; graph theory; group theory; structural stability; communication; pure mathematics; topology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue intends to provide a platform for researchers in physics and biology whose work is connected to Fibonacci (and Lucas) numbers and the golden ratio. These concepts of great mathematical simplicity and beauty continue to fascinate scholars in many fields such as quantum physics, general relativity, astronomy, chemistry, biology, and architecture, to name but a few.

In quantum physics, Fibonacci numbers and the related golden ratio have appeared in recent works devoted to nonlinear photonic crystals, quasi-one-dimensional Ising ferromagnetism and non-Abelian anyons of the Fibonacci type. They also occur in atomic physics, quasicrystals, chaos, superconductivity, astrophysics, and black holes. In biology, the golden ratio was found to appear in several works about the human body, plant phyllotaxis, DNA nucleotide sequences, the genetic code, and the topology of viruses.

The scope of this Special Issue is large enough to encompass all aspects of research in physics and biology involving the golden ratio and Fibonacci and/or Lucas numbers. We strongly encourage all our interested colleagues to submit details of their latest work.

Dr. Tidjani Negadi
Prof. Dr. Michel Planat
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Fibonacci and Lucas numbers
  • golden ratio
  • symmetry
  • quantum physics
  • DNA/RNA
  • mathematical biology
  • genetic code

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Published Papers (6 papers)

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21 pages, 2080 KiB  
Article
Algebraic Nexus of Fibonacci Forms and Two-Simplex Topology in Multicellular Morphogenesis
by William E. Butler Hoyos, Héctor Andrade Loarca, Kristopher T. Kahle, Ziv Williams, Elizabeth G. Lamb, Julio Alcántara, Thomas Bernard Kinane and Luis J. Turcio Cuevas
Symmetry 2024, 16(5), 516; https://doi.org/10.3390/sym16050516 - 24 Apr 2024
Viewed by 879
Abstract
Background: Fibonacci patterns and tubular forms both arose early in the phylogeny of multicellular organisms. Tubular forms offer the advantage of a regulated internal milieu, and Fibonacci forms may offer packing efficiencies. The underlying mechanisms behind the cellular genesis of Fibonacci and tubular [...] Read more.
Background: Fibonacci patterns and tubular forms both arose early in the phylogeny of multicellular organisms. Tubular forms offer the advantage of a regulated internal milieu, and Fibonacci forms may offer packing efficiencies. The underlying mechanisms behind the cellular genesis of Fibonacci and tubular forms remain unknown. Methods: In a multicellular organism, cells adhere to form a macrostructure and to coordinate further replication. We propose and prove simple theorems connecting cell replication and adhesion to Fibonacci forms and simplicial topology. Results: We identify some cellular and molecular properties whereby the contact inhibition of replication by adhered cells may approximate Fibonacci growth patterns. We further identify how a component 2→3 cellular multiplication step may generate a multicellular structure with some properties of a two-simplex. Tracking the homotopy of a two-simplex to a circle and to a tube, we identify some molecular and cellular growth properties consistent with the morphogenesis of tubes. We further find that circular and tubular cellular aggregates may be combinatorially favored in multicellular adhesion over flat shapes. Conclusions: We propose a correspondence between the cellular and molecular mechanisms that generate Fibonacci cell counts and those that enable tubular forms. This implies molecular and cellular arrangements that are candidates for experimental testing and may provide guidance for the synthetic biology of hollow morphologies. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
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22 pages, 574 KiB  
Article
Fibonacci-like Sequences Reveal the Genetic Code Symmetries, Also When the Amino Acids Are in a Physiological Environment
by Tidjani Négadi
Symmetry 2024, 16(3), 293; https://doi.org/10.3390/sym16030293 - 2 Mar 2024
Cited by 1 | Viewed by 2539
Abstract
In this study, we once again use a set of Fibonacci-like sequences to examine the symmetries within the genetic code. This time, our focus is on the physiological state of the amino acids, considering them as charged, in contrast to our previous work [...] Read more.
In this study, we once again use a set of Fibonacci-like sequences to examine the symmetries within the genetic code. This time, our focus is on the physiological state of the amino acids, considering them as charged, in contrast to our previous work where they were seen as neutral. In a pH environment around 7.4, there are four charged amino acids. We utilize the properties of our sequences to accurately describe the symmetries in the genetic code table. These include Rumer’s symmetry, the third-base symmetry and the “ideal” symmetry, along with the “supersymmetry” classification schemes. We also explore the special chemical structure of the amino acid proline, presenting two perspectives—shCherbak’s view and the Downes–Richardson view—which are included in the description of the above-mentioned symmetries. Our investigation also employs elementary modular arithmetic to precisely describe the chemical structure of proline, connecting the two views seamlessly. Finally, our Fibonacci-like sequences prove instrumental in quickly establishing the multiplet structure of non-standard versions of the genetic code. We illustrate this with an example, showcasing the efficiency of our method in unraveling the complex relationships within the genetic code. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
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13 pages, 311 KiB  
Article
p-Numerical Semigroups of Generalized Fibonacci Triples
by Takao Komatsu, Shanta Laishram and Pooja Punyani
Symmetry 2023, 15(4), 852; https://doi.org/10.3390/sym15040852 - 3 Apr 2023
Cited by 6 | Viewed by 1339
Abstract
For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations [...] Read more.
For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
22 pages, 17451 KiB  
Article
Voronoi Diagrams Generated by the Archimedes Spiral: Fibonacci Numbers, Chirality and Aesthetic Appeal
by Mark Frenkel, Irina Legchenkova, Nir Shvalb, Shraga Shoval and Edward Bormashenko
Symmetry 2023, 15(3), 746; https://doi.org/10.3390/sym15030746 - 17 Mar 2023
Cited by 2 | Viewed by 3004
Abstract
Voronoi mosaics inspired by seed points placed on the Archimedes Spirals are reported. Voronoi (Shannon) entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi tessellations generated by the seeds located on the Archimedes spiral and separated by linearly growing [...] Read more.
Voronoi mosaics inspired by seed points placed on the Archimedes Spirals are reported. Voronoi (Shannon) entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi tessellations generated by the seeds located on the Archimedes spiral and separated by linearly growing radial distance demonstrate a switch in their chirality. Voronoi mosaics built from cells of equal size, which are of primary importance for the decorative arts, are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal “defected” cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of seed points located on the Archimedes Spirals is reported. Voronoi tessellations generated by the phyllotaxis-inspired patterns are addressed. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
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18 pages, 380 KiB  
Article
The Metallic Ratio of Pulsating Fibonacci Sequences
by Kittipong Laipaporn, Kiattiyot Phibul and Prathomjit Khachorncharoenkul
Symmetry 2022, 14(6), 1204; https://doi.org/10.3390/sym14061204 - 10 Jun 2022
Cited by 4 | Viewed by 2169
Abstract
The golden ratio and the Fibonacci sequence (Fn) are well known, as is the fact that the ratio Fn+1Fn converges to the golden ratio for sufficiently large n. In this paper, we investigate the [...] Read more.
The golden ratio and the Fibonacci sequence (Fn) are well known, as is the fact that the ratio Fn+1Fn converges to the golden ratio for sufficiently large n. In this paper, we investigate the metallic ratio—a generalized version of the golden ratio—of pulsating Fibonacci sequences in three forms. Two of these forms are considered in the sense of pulsating recurrence relations, and their diagrams can be represented by symmetry, which is one of their distinguishing characteristics. The third form is the Fibonacci sequence in bipolar quantum linear algebra (BQLA), which also pulsates. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
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10 pages, 1852 KiB  
Article
DNA Structure and the Golden Ratio Revisited
by Stuart Henry Larsen
Symmetry 2021, 13(10), 1949; https://doi.org/10.3390/sym13101949 - 16 Oct 2021
Cited by 4 | Viewed by 13671
Abstract
B-DNA, the informational molecule for life on earth, appears to contain ratios structured around the irrational number 1.618…, often known as the “golden ratio”. This occurs in the ratio of the length:width of one turn of the helix; the ratio of the spacing [...] Read more.
B-DNA, the informational molecule for life on earth, appears to contain ratios structured around the irrational number 1.618…, often known as the “golden ratio”. This occurs in the ratio of the length:width of one turn of the helix; the ratio of the spacing of the two helices; and in the axial structure of the molecule which has ten-fold rotational symmetry. That this occurs in the information-carrying molecule for life is unexpected, and suggests the action of some process. What this process might be is unclear, but it is central to any understanding of the formation of DNA, and so life. Full article
(This article belongs to the Special Issue Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology)
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