Related papers: A New Theorem Relating the Tangent Secant Theorem …
Much has been written about the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ and this strange number appears mysteriously in many mathematical calculations. In this article, we review the appearance of this number in the graph theory. More…
The golden ratio and Fibonacci numbers are found to occur in various aspects of nature. We discuss the occurrence of this ratio in an interesting physical problem concerning center of masses in two dimensions. The result is shown to be…
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…
We find an interesting relationship between the golden ratio, the Moebius function, the Euler totient function and the natural logarithm - central players in the theory of numbers. A number of identities involving the golden ratio and its…
We establish some new constructions of the golden ratio in an arbitrary triangle using symmedians and nine-point circle.
Exploiting Markoff's Theory for rational approximations of real numbers, we explicitly link how hard it is to approximate a given number to an idealized notion of growth capacity for plants which we express as a modular invariant function…
There are useful and useless golden ratios. The useful one helps in traffic. The useless and rather mysterious one arises in hydrodynamics of point vortices, which we discuss in detail.
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…
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…
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…
We consider a variational formalism to describe black holes solution in higher dimensions. Our procedure clarifies the arbitrariness of the radius parameter and, in particular, the meaning of the event horizon of a black hole. Moreover, our…
The concept of the Quantum Ratio was born out of the efforts to find a simple but universal criterion if the center of mass (CM) of an isolated (microscopic or macroscopic) body behaves quantum mechanically or classically, and under which…
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…
Based on artistic interpretations, art professor Christopher Bartlett (Towson University, USA) suggested introducing a new mathematical constant related to the golden ratio. He called it the chi ratio, as chi follows phi in the Greek…
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 apparent gap between the measured and the expected value for the semileptonic branching ratio of $B$ mesons has become more serious over the last year. This is due to the improved quality of the data and to the increasing maturity of…
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…
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…
One central study that constitutes a major branch of quantum resource theory is the hierarchy of states. This provides a broad understanding of resourcefulness in certain tasks in terms of efficiency. Here, we investigate the maximal…
We prove a Goldstone Theorem in thermal relativistic quantum field theory, which relates spontaneous symmetry breaking to the rate of space-like decay of the two-point function. The critical rate of fall-off coincides with that of the…