Related papers: Substitutions for tilings $\{p,q\}$

Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored…

A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…

A particular orthogonal map on a finite dimensional real quadratic vector space (V,Q) with a non-degenerate quadratic form Q of any signature (p,q) is considered. It can be viewed as a correlation of the vector space that leads to a dual…

In this paper, we derive eight basic identities of symmetry in three variables related to $q$-Euler polynomials and the $q$-analogue of alternating power sums. These and most of their corollaries are new, since there have been results only…

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields.…

The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory…

Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that…

In this paper, we connect two types of representations of a permutation $\sigma$ of the finite field $\F_q$. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and $k$ copies of…

A quasiperiodic 7-fold rhombic tiling is constructed with an iterative substitution scheme. The inflation factor is 5.04892..., the square of the longer diagonal of a regular heptagon. There are many substitutions possible that fill larger…

We give a general construction leading to different non-isomorphic families $\Gamma_{n,q}(\K)$ of connected $q$-regular semisymmetric graphs of order $2q^{n+1}$ embedded in $\PG(n+1,q)$, for a prime power $q=p^h$, using the linear…

By considering the tiling of an $N$-board (a linear array of $N$ square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci…

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine…

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the…

The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…

A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and…

The Fibonomial coefficients are well-known analogues of the classical binomial coefficients. In 2009, Sagan and Savage introduced a combinatorial interpretation for these coefficients, based on tiling a rectangular grid. More recently,…

This paper presents the construction and various properties of substitution tilings related to Stern's diatomic series and based on tiles decorated with elements of $\mathbb{F}_p$ for some odd prime number $p$. These substitution tilings…

The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\mathbb{1}_F * \mathbb{1}_A = 1$, there is also a periodic…

We use the subgraph replacement method to prove a simple product formula for the tilings of an 8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999),…