Related papers: Noncongruent equidissections of the plane
We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and…
One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be…
We give a proof of Ollinger's conjecture that the problem of tiling the plane with translated copies of a set of $8$ polyominoes is undecidable. The techniques employed in our proof include a different orientation for simulating the Wang…
We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…
We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean…
Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinite towers of orbifolds uniformized by the complex 2-ball and some orbifolds over K3 surfaces uniformized by the 2-ball. Corresponding…
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…
We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular…
A survey of tilings in the plane for a general audience.
In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the…
In this paper we study geometric coincidence problems in the spirit of the following problems by B. Gr\"unbaum: How many affine diameters of a convex body in $\mathbb R^n$ must have a common point? How many centers (in some sense) of…
We show that if $X$ is an arc-like continuum, then for every point $x \in X$ there is a plane embedding of $X$ in which $x$ is an accessible point. This answers a question posed by Nadler in 1972, which has become known as the Nadler-Quinn…
Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.
We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce…
Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points.…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
The concepts of symmetry and its breakdown are investigated in two different terms according to whether the resulting asymmetry is universal or only obtained for a special configuration: we shall illustrate this by considering in the first…
Plane triangulations with all vertices of degree $3$ or $6$ are enumerated. A plane triangulation is said to be akempic if it has a $4$-colouring such that no two adjacent triangles have the same three colours and this colouring is not…