Related papers: Substitution drawing rules on the Fibonacci word
This is my (reviewed) PhD manuscript. It contains 6 Chapters, which contains mostly already published work, except for Chapter 5 which is new. Chapter 1 introduce basic notions on fractal geometry: the Fourier dimension, the thermodynamical…
This work explores new arithmetic and combinatorial structures arising from the interplay between Farey-type graphs, Fibonacci expansions, and operadic constructions. We introduce Fibonadic numbers, defined as an inverse limit under the…
This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
Trees in works of art have stirred emotions in viewers for millennia. Leonardo da Vinci described geometric proportions in trees to provide both guidelines for painting and insights into tree form and function. Da Vinci's Rule of trees…
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The…
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The…
The sequence of partial sums of Fibonacci numbers, beginning with $2$, $4$, $7$, $12$, $20$, $33,\dots$, has several combinatorial interpretations (OEIS A000071). For instance, the $n$-th term in this sequence is the number of length-$n$…
The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary…
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are…
The Fibonacci word is the fixed point beginning with the letter $a$ of morphism $\sigma(a)=ab$, $\sigma(b)=a$ defined over the alphabet $\{a,b\}$. In this paper, we get explicit expression of the number of distinct fractional powers in each…
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
This work presents a detailed analytical and geometrical investigation of the (2+1)-dimensional Boiti-Leon-Pempinelli system, a nonlinear dispersive model arising in the context of fluid and plasma dynamics. By employing a projective…
Quasicrystals are fractal due to their self similar property. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
We explicitly construct fractals of dimension 4-epsilon on which dimensional regularization approximates scalar-field-only quantum-field-theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and…
We present a general scheme how to construct a substitution rule for generating $d$-dimensional analogues of the paperfolding structures. This substitution is proven to be primitive, so that the translation action on the hull forms a…
We define the notion of circular words, then consider on such words a constraint derived from the Fibonacci condition. We give several results on the structure of these circular words, then mention possible applications to various…
Assuming a fractal distribution of matter in the universe, consequences that follow from the General Theory of Relativity and the Copernican Principle for fractal cosmology are examined. The change in perspective necessary to deal with a…