The intimate relationship between the Penrose and the Taylor-Socolar tilings is studied, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship and offers straightforward proofs of their basic properties.

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonome...

We find all convex lattice polygons in the plane (up to equivalence) with at most one interior lattice point. A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. In 1980, Arkinstall [1] proved that, up to lattice equivalence (defined belo...

STATEMENT OF PROBLEM Although the implant-abutment connection may prevent crestal bone loss around dental implants, its failure often leads to treatment failure. Microgap and micromovement of the implant-abutment connection could be causes of bone resorption around dental implant neck. PURPOSE The purpose of this study was to compare torque resistance and microgaps between a new cone and inde...

The role of opposition in rough set theory is laid bare. There are two sources which generate oppositions in rough sets: approximations and relations. In the former case, we outline a hexagon and a cube of oppositions. In the second case, we define a classical square of oppositions and also a tetrahedron when considering the standpoint of two agents.

A new scheme for spontaneous formation of hexagonal optical patterns in photorefractive crystals with dominant reflection gratings and a single pump wave is proposed and realized experimentally. It is shown theoretically that spontaneous hexagon formation results from random small-angle scattering of the pump wave inside the crystal. The threshold of spontaneous hexagon formation is found.

We compute the number of rhombus tilings of a hexagon with sides n, n, N , n, n, N , where two triangles on the symmetry axis touching in one vertex are removed. The case of the common vertex being the center of the hexagon solves a problem posed by Propp.

We study color confinement properties of the multi-instanton system, which seems to carry an essence of the nonperturbative QCD vacuum. Here we assume that the multi-instanton system is characterized by the infrared suppression of instantons as f (ρ) ∼ ρ −5 for large size ρ. We first investigate a monopole-clustering appearing in the maximally abelian (MA) gauge by considering the correspondenc...

A study of the implant to abutment connection is of paramount importance as it is the primary determinant of the strength and stability of the implant supported restoration, which in turn determines the restoration’s prosthetic stability. Traditionally, the Branemark's external hexagon has been widely used but its significant complications like abutment screw loosening, rotational misfit at imp...