The Golden Ratio in Nature: A Tour across Length Scales
Round 1
Reviewer 1 Report
Comments for author File: 
Comments.pdf
Author Response
We thank the reviewer very much for their valuable comments and interest in the manuscript.
We too are interested in the connections between Phi and fractals and non-linear dynamics, and thank the reviewer for the links to additional works. We have included the Vieta nested root approximation (Equation 18), the two references, and the connection to fractal behaviour and non-linear dynamics, such as those demonstrated in the mechanical models of Lindner (addition of a paragraph from 170).
We have corrected to two typographical errors as noted by the reviewer (now lines 229 and 472), and a few more that we have also noted.
Reviewer 2 Report
Review Report of
“THE GOLDEN RATIO IN NATURE: A TOUR ACROSS LENGTH SCALES”
This review paper presents the research on the mathematical properties of the Golden ratio. The work investigates where in nature this proportion can be found. The authors perform the analysis of the Golden proportion from the astronomical to atomic scales. Throughout the whole paper, the mathematical aspect of this phenomenon has been analyzed.
It is thoroughly illustrated how prevalent this single number is within the natural universe. By all means, this is an interesting review study and the authors have pointed out certain interesting properties of the Golden number. The paper is generally well written and structured. It is concise and with a very clear point.
Conclusion
I recommend the paper for publication since, in my opinion, it fulfills all the criteria and standards to be in your journal. In my belief, the research should be published in its present form. This topic may be potentially attractive to the readers of Symmetry, especially because of its mathematical aspect.
Author Response
We thank the reviewer for their comments.
We have corrected the typographical errors that we have spotted and the two that were noted by another reviewer, and amended the grammar where necessary.
Reviewer 3 Report
The scientific novelty presented in the article and its impact on the development of science is insignificant, even that the presentation is nice. Actually, this article is a survey about golden ratio.
This topic is an ultra-studied topic over time and cannot be an article to be published in a scientific journal like Symmetry, but in a journal dedicated to high school students or in the classes of the College Math Circles or high school.
The presentation begins with a brief history of the golden ratio. They write its development in the continued fraction and remember and remember about the approximation of rational numbers. Then, the domains where the golden ratio can be found are presented: Spiral Galaxies, The Logarithmic Spiral, Variable Stars, Planetary Orbits, the KAM (Kolmogorov–Arnold–Moser) theorem, Phyllotaxis, The Ultimatum Game, The Human Body, Proteins and Amino Acids, Penrose Tilings and Quasicrystals, Atomic Bond Lengths, Black Holes, Quantum Gravity and E8.
The impressive number of references proves that this can be a bachelor's or master's thesis.
Author Response
We thank the reviewer for their time reading and providing comments on the manuscript. We were disappointed to read the view that the manuscript has an insignificant impact on the development of science, and respectfully disagree. It is true that the Golden ratio is a highly studied topic; indeed such is the interest and relevance across different fields of science that we selected this “Fibonacci and Lucas Numbers and the Golden Ratio in Physics and Biology” Special Edition in Symmetry to publish this work.
The significant interest in the apparent observation of Phi in nature has led to a number of misconceptions, misinterpretations and incorrect assumptions. Here, as a review, we aim to tour through the physical and natural world, from galactic to sub-atomic length scales, highlighting where Phi has been noted, and to describe any theoretical basis for its presence at each stage. A common misattribution, for example, that we highlight is in the logarithmic spiral of the Nautilus shell.
We have made a number of additions to the manuscript to reinforce the findings. From line 132, we have added the paragraph:
The purpose of this paper is to discuss a selection of these phenomena and to assess the evidence suggesting a link to the Golden ratio. For the purposes of this study, a natural phenomenon is considered to be linked to the Golden ratio if two conditions are satisfied: the existence of observational or experimental evidence of φ, and a rigorous theoretical justification explaining its presence. Where one exists without the other, any evidence shall be considered to be inconclusive.
And each subsection now:
… ends by summarising the evidence for a link to the Golden ratio, concluding for each case whether such a link exists according to the criteria given above.
Summary paragraphs for each section are on lines (Spiral Galaxies) 126 - 130; (Variable Stars) 180 – 183; (KAM (Planetary Orbits)) 242 – 246; (Phyllotaxis) 264 – 266; (Ultimatum Game) 292 – 294; (Human Body (Heart, Brain)) 321 – 325; (Proteins; Amino Acids*) 364 – 370; (Quasicrystals) 425 – 426; (Atomic Bond Lengths) 445 – 448; (Black Holes; Quantum Gravity; E8) 480 – 483 and 498 – 500;
*For the analysis of protein and amino acid shape, we have added a statistical analysis (following a method described by Santos et al. [50]) detailed in lines 335 – 345.
We hope that the further analysis and statements, supported by additional references 27, 28, 29, 37, 43, 44, 47, 50 and 51 showing where evidence for the presence of Ï• comes from, will address the reviewer’s concerns over publication of this manuscript in the Special Edition of Symmetry.
Round 2
Reviewer 3 Report
My opinion remains the same.
In 22 pages of the article there are 19 mathematical formulas, and all of these are general formulas.
If the Editor considers that it is an article worthy of publication in Symmetry, then I have nothing against it.
