An integer representation for periodic tilings of the plane by regular polygons

We describe a representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely by a (2+n)×4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. We discuss several properties of this representation and describe how to exploit the representation elegantly and efficiently for reconstruction, rendering, and automatic crystallographic classification by symmetry detection

Konstrukcija D-grafova kod periodičkih popločavanja

This paper presents an algorithm which allows to derive classification methods concerning periodic tilings in any dimension, theoretically. By the help of this, one can check former results of the topic (e. g. [BM98, BM00]). An implementation of the algorithm yields the complete enumeration of non-isomorphic three-dimensional D-graphs with 5 elements, given as illustration.U radu je dan algoritam koji teoretski omogućuje izvođenje metoda za klasifikaciju periodičkih popločavanja u svakoj dimenziji. Pomoću toga se može provjeriti raniji rezulatat dan u radu (e. g. [BM98, BM00]). Primjenom algoritma prikazana je potpuna klasifikacija neizomorfnih trodimenzionalnih D-grafova s 5 elemenata

Heesch numbers of unmarked polyforms

A shape's Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of nontiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds