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Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples…
There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon…
We have pursued in the literature a fractal-like structure for the fractional quantum Halll effect-FQHE which consider the Hausdorff dimension associated with the quantum mechanics paths and the spin of the particles or quasiparticles…
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is…
We discuss a variation of Takagi curves based, however, more on algebraic than geometric principles. Namely, we construct functions of loops in a special binary representation. The graph of these functions usually has chaotic and fractal…
This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its…
In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we showed that for each…
In this manuscript, by using Belyi maps and dessin d'enfants, we construct some concrete examples of Strebel differentials with four double poles on the Riemann sphere. As an application, we could give some explicit cone spherical metrics…
A fractal approach to numerical analysis of electromagnetic space-time crystals, created by three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency, is presented. Finite models of electromagnetic…
A method is described for calculating the approximate fractal dimension from a set of N values y sampled from a waveform between time zero and t. The waveform was subjected to a double linear transformation that maps it into a unit square.
Two proofs are provided that the narrow ideal class group of squared ideals of a quadratic number field is isomorphic to the kernel of a homomorphism between cohomology groups for Pell conics, Lemmermeyer's obstruction to descent for Pell…
We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.
We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.
We perform fractal analysis on a closed classical hard-wall billiard, the circular billiard with a straight cut, assuming there are two openings on the boundary. We use a two-dimensional set of initial conditions that produce all possible…
Generalizations of Bell polynomials, Bell numbers, and Stirling numbers of the second kind have been introduced and their generating functions were evaluated.
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.
A new method Spherical Rectangular Equal-Area Grid (SREAG) was proposed in Malkin (2019) for splitting spherical surface into equal-area rectangular cells. In this work, some more detailed features of SREAG are presented. The maximum number…
A fractal construction shows that, for any beta>0, the beta-skeleton of a point set can have arbitrarily large dilation. In particular this applies to the Gabriel graph.
We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and…