Related papers: Unavoidable golden ratio
Many algorithms are specified with respect to a fixed but unspecified parameter. Examples of this are especially common in cryptography, where protocols often feature a security parameter such as the bit length of a secret key. Our aim is…
It is known that the golden ratio $\alpha =\frac{1+\sqrt{5}}{2}$ has many applications in geometry. In this paper we consider some geometric properties of finite Blaschke products related to the golden ratio.
We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the…
Numerical analysis indicates that there exists an unexpected new ordered chaos for the bounded one-dimensional multibarrier potential. For certain values of the number of barriers, repeated identical forms (periods) of the wavepackets…
Fractional parts of the first $N$ natural numbers fill the unit interval with asymptotically uniform density. However, the gaps around rational points shrink at an asymptotically lower rate $N^{-1/2}$, and their widths scale with the Thomae…
One source of beauty in mathematics is totally unexpected connections between two fundamentally different objects. For instance, is it not surprising that the time period of a real simple pendulum is linked with a function arising out of…
In principle, the combined measurements of the mass and radius a giant exoplanet allow one to determine the relative fraction of hydrogen and helium and of heavy elements in the planet. However, uncertainties on the underlying physics imply…
The fundamental laws of physics are time-symmetric, but our macroscopic experience contradicts this. The time reversibility paradox is partly a consequence of the unpredictability of Newton's equations of motion. We measure the dependence…
High entropy alloys add a new dimension, atomic-scale randomness and the associated scale-dependent composition fluctuations, to the traditional metallurgical axes of time-temperature-composition-microstructure. Alloy performance is…
A ternary complex tree related to the golden ratio is used to show how the theory of complex trees works. We use the topological set of this tree to obtain a parametric family of trees in one complex variable. Even though some real ferns…
The need for improved functionalities is driving the search for more complicated multi-component materials. Despite the factorially increasing composition space, ordered compounds with 4 or more species are rare. Here, we unveil the…
Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for…
Quasi-crystals are aperiodic structures that present crystallographic properties which are not compatible with that of a single unit cell. Their revolutionary discovery in a metallic alloy, less than three decades ago, has required a full…
We prove that the golden angle (an angle that divides the circle in the golden ratio) is not constructible using straightedge and compass.
We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross…
In this work I look at the distribution of primes by calculation of an infinite number of intersections. For this I use the set of all numbers which are not elements of a certain times table in each case. I am able to show that it exists a…
A new class of self-similar ordered structures with non-crystallographic point symmetries is presented. Each of these structures, named superquasicrystals, is given as a section of a higher-dimensional "crystal" with recursive superlattice…
Local convergence techniques have become a key methodology to study sparse random graphs. However, convergence of many random graph properties does not directly follow from local convergence. A notable, and important, such random graph…
Stellar chemical element ratios have well-defined systematic trends as a function of abundance, with an excellent correlation of these trends with stellar populations defined kinematically. This is remarkable, and has significant…
Modeling the spontaneous evolution of morphology in natural systems and its preservation by proportionate growth remains a major scientific challenge. Yet, it is conceivable that if the basic mechanisms of growth and the coupled kinetic…