Related papers: On the Sierpinski triangle and its generalizations
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…
Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken…
Generalised Sierpinski carpets are planar sets that generalise the well-known Sierpinski carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the…
Under certain continuity conditions, we estimate upper and lower box dimension of graph of a function defined on the Sierpinski gasket. We also give an upper bound for Hausdorff dimension and box dimension of graph of function having finite…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
The embedding of a given point set with non-crystallographic symmetry into higher-dimensional space is reviewed, with special emphasis on the Minkowski embedding known from number theory. This is a natural choice that does not require an a…
We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology…
Hilbert space fragmentation refers to exponential growth in the number of dynamically disconnected Krylov sectors with system size. It is taken as evidence of ergodicity breaking, since conventional symmetries generate at most a polynomial…
The term fractal describes a class of complex structures exhibiting self-similarity across different scales. Fractal patterns can be created by using various techniques such as finite subdivision rules and iterated function systems. In this…
The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any $d>3$ we find Zariski…
We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…
Research done during the previous century established our Standard Cosmological Model. There are many details still to be filled in, but few would seriously doubt the basic premise. Past surveys have revealed that the large-scale…
This article gives a summary of the author's Ph.D. dissertation (arXiv:1609.06297). In addition to an overview of notions and results, it also provides sketches of various proofs and simplified presentations of certain abstract results of…
In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical…
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…
The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.
The permutation associated with the decimal expression of the binary reflected Gray code with $N$ bits is considered. Its cycle structure is studied. Considered as a set of points, its self-similarity is pointed out. As a fractal, it is…
It is shown how derived brackets naturally arise in sigma-models via Poisson- or antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression…
We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often…
The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical…