Regular production systems and triangle tilings

Regular production systems and triangle tilings

We discuss regular production systems as a tool for analyzing tilings in general. As an application we give necessary and sufficient conditions for a generic triangle to admit a tiling of H and show that almost every triangle that admits a tiling is “weakly aperiodic.” We pause for a variety of other applications, such as non-quasi-isometric maps between regular tilings, aperiodic Archimedean t...

Regular Production Systems and Tilings in the Hyperbolic Plane

Further Triangle tilings

Here we discuss which triangles do, and which don’t, admit a tiling of H,E, and S. These notes are meant to pick up as “Regular Production Systems and Triangle Tilings” [16] leaves off. We pause for a conjecture and some additional nomenclature: Let T be any triangle in X = H,E, S. A configuration by T is a collection of congruent copies of T, for each pair of which meet edge-to-edge and vertex...

Combinatorially Regular Polyomino Tilings

Let T be a regular tiling of R which has the origin 0 as a vertex, and suppose that φ : R → R is a homeomorphism such that i) φ(0) = 0, ii) the image under φ of each tile of T is a union of tiles of T , and iii) the images under φ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertice...

Transitive triangle tilings in oriented graphs

In this paper, we prove an analogue of Corrádi and Hajnal's result. There exists n 0 such that for every n ∈ 3Z when n ≥ n 0 the following holds. If G is an oriented graph on n vertices and δ 0 (G) ≥ 7n/18, then G contains a perfect T T 3-tiling, which is a collection of vertex disjoint transitive triangles covering every vertex of G. This result is best possible, as there exists an oriented gr...