Related papers: On the Fibonacci tiling and its modern ramificatio…
Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define - perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which…
We provide a new perspective on the divisor theory of graphs, using additive combinatorics. As a test case for this perspective, we compute the gonality of certain families of outerplanar graphs, specifically the strip graphs. The Jacobians…
The existence of cycles of the matrix maps in Fibonacci class of lattices is well established. We show that such cycles are intimately connected with the presence of interesting positional correlations among the constituent `atoms' in a one…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of $a^3b$-quadrilaterals with some irrational angle: there are a sequence of…
Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three…
We develop the basic and new tools for classifying non-side-to-side tilings of the sphere by congruent triangles. Then we prove that, if the triangle has any irrational angle in degree, such tilings are: a sequence of 1-parameter families…
Great part of the interest in complex networks has been motivated by the presence of structured, frequently non-uniform, connectivity. Because diverse connectivity patterns tend to result in distinct network dynamics, and also because they…
We develop a new class of clockwork theories with an augmented structure of the near-neighbour interactions along a one-dimensional closed chain. Such a topology leads to new and attractive features in addition to generating light states…
Convergence theory is an extension of general topology. In contrast with topology, it is closed under some important operations, like exponentiation. With all its advantages, convergence theory remains rather unknown. It is an aim of this…
We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they…
The Fibonacci numbers are familiar to all of us. They appear unexpectedly often in mathematics, so much there is an entire journal and a sequence of conferences dedicated to their study. However, there is also another sequence of numbers…
This paper discusses the formalization of proofs "by diagram chasing", a standard technique for proving properties in abelian categories. We discuss how the essence of diagram chases can be captured by a simple many-sorted first-order…
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…
This paper presents a more complete version than hitherto published of our explanation of a transition from regular to irregular motions and more generally of the nature of a certain kind of deterministic chaos. To this end we introduced a…
Recently, Engel et al. discussed phonon broadening as observed in 3D quasicrystals on the basis of calculations on the Fibonacci chain. We show that the paper contains several statements and assumptions that are contradicted by factual…
Thus far, in spite of many interesting developments, the overall progress towards a systematic study and classification of various 'strange' metallic states of matter has been rather limited. To that end, it was argued that a recent…
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing…
The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. The main result is twofold: (1) we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the…