Narrow Results

A construction for the generation of eight-fold symmetry planar patterns is introduced. The basic building blocks are four triangle prototiles with six edge lengths. Subpatterns with three prototile shapes and two edge lengths can also be derived.

In a recent paper several species of octagonal patterns have been introduced with the help of a construction which allows us to derive them by means of inflation rules. Non-deterministic patterns can be generated by composition of the inflation rules. In this paper we show how a similar construction produces patterns with hexagonal symmetry. The non-deterministic rhombus–triangle tilings are obtained by local rearrangements of tiles which are included in the inflation rules. This property allows to compute the configurational entropy.

An interpretation of Icosahedral Danzer tilings in terms of algebraic substitutions is used in order to study the Fourier transform of suitable mass distributions. Numerical results are obtained for a mass distribution placed on vertex positions.