Regular production systems and triangle tilings

Spherical F-Tilings by Triangles and r-Sided Regular Polygons, r >= 5

The study of dihedral f-tilings of the sphere S2 by spherical triangles and equiangular spherical quadrangles (which includes the case of 4-sided regular polygons) was presented in [3]. Also, in [6], the study of dihedral f-tilings of S2 whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle was described (we believe that the analysis considering scal...

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...

A Fractal Fundamental Domain with 12-fold Symmetry

Square triangle tilings are relevant models for quasicrystals. We introduce a new self-similar tile-substitution which yields the well-known nonperiodic square triangle tilings of Schlottmann. It is shown that the new tilings are locally derivable from Schlottmann’s, but not vice versa, and that they are mutually locally derivable with the undecorated square triangle tilings. Furthermore, the r...

Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

The study of dihedral f-tilings of the Euclidean sphere S2 by triangles and rsided regular polygons was initiated in 2004 where the case r = 4 was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and r-sided regular polygons, for any r ≥ 5, was described. Later on, in [3], the classification of all f-tilings of S2 whose prototiles are an equilateral t...

Regular Production Systems and Tilings in the Hyperbolic Plane