Related papers: Balancing on the edge, the golden ratio, the Fibon…
Fibonacci numbers and the golden ratio can be found in nearly all domains of Science, appearing when self-organization processes are at play and/or expressing minimum energy configurations. Several non-exhaustive examples are given in…
The problem of the universal form of the size spectrum is analyzed. The half-widths of two wings of spectrum is introduced and it is shown that their ratio is very close to the golden fraction. In appendix it is shown that behind the golden…
It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic…
This paper explores the Fibonacci sequence and the Golden Ratio as organizing principles for visual composition and abstraction in painting. The author shows how recursive proportional systems, long associated with natural growth and…
In this thesis we examined mathematical properties of Fibonacci numbers and applications of this numbers in the nature,geometry and economy.We obtained Golden section and proved some mathematical identities using Golden section. Infinity of…
An amusing connection between Ford circles, Fibonacci numbers, and golden ratio is shown. Namely, certain tangency points of Ford circles are concyclic and involve Fibonacci numbers. They form four circles that cut the x-axis at points…
It is demonstrated that iterative repeating of some simple geometric construction leads unavoidably in the limit to the golden ratio. The procedure appears to be quickly convergent regardless of a ratio of initial elements sizes. This could…
The Fibonacci sequence is a series of positive integers in which, starting from $0$ and $1$, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio.…
In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the…
We find various series that involves the central binomial coefficients $\binom{2n}{n}$, harmonic numbers and Fibonacci Numbers.\\ Contrary to the traditional hypergeometric function $_pF_q$ approach, our method utilizes a straightforward…
We write out relations between the base of natural logarithms ($e$), the ratio of the circumference of a circle to its diameter ($\pi$), the golden ratios ($\Phi_p$) of the additive $p$-sequences, and the ratio of the diagonal of a square…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
We derive interesting arctangent identities involving the golden ratio, Fibonacci numbers and Lucas numbers. Binary BBP-type formulas for the arctangents of certain odd powers of the golden ratio are also derived, for the first time in the…
The Fibonacci sequence (FS) possesses exceptional mathematical properties that have captivated mathematicians, scientists, and artists across centuries. Its intriguing nature lies in its profound connection to the golden ratio, as well as…
As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (1 + sqrt(5))/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root eta of X^3 - X^2 - X -…
Expansions in the Golden ratio base have been studied since a pioneering paper of R\'enyi more than sixty years ago. We introduce closely related expansions of a new type, based on the Fibonacci sequence, and we show that in some sense they…
Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver…
In this work, we have abstractly generalized the similarity law for multidimensional vectors. Initially, the law of similarity was derived for one-dimensional vectors. Although it operated with such values of the ratio of parts of the…