Related papers: Fibonacci words in hyperbolic Pascal triangles
A new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced in [H.B, L.N, L.Sz: Hyperbolic Pascal triangles]. The mathematical background goes back to the regular mosaics in the hyperbolic plane. The…
Recently, a new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced. The mathematical background goes back to the regular mosaics in the hyperbolic plane. In this article, we investigate the paths…
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the…
In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose…
In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to $\{4,q\}$ with $q\ge5$. The method is based on the theory of linear recurrences, and the results…
The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…
In this paper we study the word problem of groups corresponding to tessellations of the hyperbolic plane. In particular using the Fibonacci technology developed by the second author we show that groups corresponding to the pentagrid or the…
We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a…
In this paper we introduce a new type of Pascal's pyramids. The new object is called hyperbolic Pascal pyramid since the mathematical background goes back to the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The definition…
We give an overview about some elementary properties of Hoggatt matrices, which are generalizations of Pascal triangle, and study q-analogs and Fibonacci analogs and derive a common generalization.
We identify the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $4$ and and element of order $p\geq 2$. Thus $\Gamma$ has a…
The paper explores combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. These infinite sequences, measures the diversity of subwords in Fibonacci words, showing non-decreasing…
The aim of this paper is to determine the elements which are in two pairs of sequences linked to the regular mosaics $\{4,5\}$ and $\{p,q\}$ on the hyperbolic plane. The problem leads to the solution of diophantine equations of certain…
Pascal's triangle will give the number of geodesics from the identity to each point of ${\bf Z}^2$ if you write it in each of the quadrants. Given a group $G$ and generating set $\cal G$ we take the {\it Pascal's function} $p_{\cal G}: G…
Fibonacci word fractals are a class of fractals that have been studied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which possesses…
The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition.…
In this second paper, we look at the following question: are the properties of the trees associated to the tilings $\{p,4\}$ and $\{p$+$2,3\}$ of the hyperbolic plane still true if we consider a finitely generated tree by the same rules but…
We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension…
In two previous papers we have presented partition formulae for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree.…
We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps…