A few references to Fibonacci numbers, 1,1,2,3,5,8,13,21,34,55,89… within this website:
What do any of us really know? Are things always simple before becoming complex? It seems to be true, all except around those issues coming out of the big bang theory as promoted by Hawking, Guth, and many others. Now, more recently Princeton’s Paul Steinhardt and his coterie have been getting some traction with their quest for a new kind of matter. Dan Shechtman (wiki) before him looked at the small-scale and got his Nobel prize in Chemistry in 2011 for his discovery of quasicrystals, the imperfect geometries based on five tetrahedrons sharing a common edge and lacking a translational symmetry. In our book, it is confined by notation and its respective Fibonacci sequences. Of course, outside of the 202 notations, that statement is meaningless or gobbledegook. The bottomline is arrogance impedes ideas and creativity. Remember Fairbairn‘s comment about “novel approaches.”
Scale Invariant Sphere Dynamics. From the infinitesimal sphere to the movement of galaxies, pi and phi (circles and Fibonacci sequences), are fundamental dynamics within everything. Pi crosses notations; phi builds within a given notation. This model not only uses numbers and geometries, it uses pi, phi, prime numbers, values, and more where big bang cosmology is based on singularities that do not account for dimensionless constants like pi. The mathematics of materialism generally disregards other systems of engagement. How is it that pi is scale invariant? What are the deep dynamics of spheres? We are trying to learn… we are in the earliest stages of our studies of the Fourier transforms and integral transforms. Of course, we’d welcome any-and-all help to understand these disciplines as well as Steven Strogatz does.
Aristotle’s Mistake: In 2015, my life changed because I came upon a reference to an article titled, “Mysteries in Packing Regular Tetrahedra.” That article amazed me. It took over 1800 years to catch Aristotle’s mistake. Yet, along that way, Averroës (Abu al-Walid Mohammad ibn Ahmad al Rushd (1126–1198), Leonardo of Pisa (Fibonacci) (c. 1228), Roger Bacon (c. 1214–1294), and Thomas Aquinas (c. 1225–1274) were among the greats of their time who reinforced his mistake. As a result, none of them would ever know about a most fundamental geometric gap. First, inferred by Johannes Müller von Königsberg (1436–1476), then documented in 1480 by Paul of Middelburg, a professor of astrology in Padua, the discussion was re-birthed by Dirk Struik (MIT) in 1926 while studying in Rome. Most recently, in December 2012, Jeffrey C. Lagarias and Chuanming Zong [also see, May 2020] brought it to life again. Yet, none of these people in their time contemplating that gap ever thought that it just might opened a path to quantum fluctuations, indeterminacy, and imperfection. Such a highly-speculative statement would appear to most physicists today to be uninformed. I do not believe that I would be overplaying my hand to say that this gap makes us all equally human. It is the beginnings of all our imperfections.
What could be more basic than waves, particles and fluctuations? How do we relate algebraic geometries (Grothendieck‘s scheme theory), Euclidean geometries, projective geometry, category theory, Mandelbrot set, Julia set, Möbius transformations, Kleinian group, S-matrix theory, unitarity equations, Hermitian analyticity, Golden ratio (Phi), the Fibonacci sequence, fluctuation theory, ratio analysis, pi, cubic-close packing of equal spheres, ring theory, and lattice generation? Out of that group we settled on pi and cubic close-packing of equal spheres at the Planck scale to begin.
The first ten notations. There is so much to learn and so little time (our life is short). I’ll be working on this 1-202 chart until I die. Notwithstanding, the Buckingham π theorem is part of our understanding of homogeneity and isotropy; it’s the heart of dimensional analysis. The Buckingham π theorem takes π to the next level along with the pure numbers, i, e, and φ. It is a thrust in the intellectual direction that ratios and equations are real and the things of space and time are derivative.
So, yes, we will continue our studies of those pure numbers, i, e, and φ and de Moivre numbers, as part of our work on the theory of group characters, and the discrete Fourier transform seeking to justify our belief that all these expressions of mathematics are inherent and active at the Planck scale.
Eventually we want to discuss how these 64 notations open up discussions of concepts within algebraic geometries, projective geometry, Euclidean geometries, category theory, Mandelbrot sets, Julia sets, Möbius transformations, Kleinian groups, S-matrix theory, unitarity, bootstrapping, Hermitian analyticity, the Golden ratio (Phi), the Fibonacci sequence, fluctuation theory, ratio analysis, and ring theory.
Qisheng Lin and John D. Corbett, “New Building Blocks…” “According to higher dimensional projection methods, a series of cubic ACs (approximant crystals) exist with orders (q/p) denoted by any two consecutive Fibonacci numbers, i.e., q/p = 1/1, 2/1, 3/2, 5/3 … F n+1/F n (1).” http://www.pnas.org/content/103/37/13589.full