6-Fold Quasiperiodic Tilings With Two Diamond Shapes

TL;DR

This paper introduces a method for creating 6-fold quasiperiodic tilings using two diamond-shaped tiles with specific matching rules, enabling indefinite inflation and complex pattern formation.

Contribution

It presents two novel aperiodic inflation schemes that generate large quasiperiodic tilings from just two diamond shapes with matching rules.

Findings

Enables indefinite inflation of quasiperiodic patterns

Creates complex Escher-like figures through tile deformation

Achieves 6-fold rotational symmetry in tilings

Abstract

A set of tiles for covering a surface is composed of two types of tiles. The base shape of each one of them is a diamond or rhombus, both with angles 60 and 120 degrees. They are distinguished by labeling one as an acute diamond with a base angle of 60 degrees, the other one as an obtuse diamond with a base angle of 120 degrees. The two types of tiles can be marked with arrows, notches, or colored lines to keep them distinct. Notches can be used as matching rules such that the acute diamonds can form a star with 6-fold rotational symmetry among other matches. Similarly, three obtuse diamonds can be matched with 3-fold rotational symmetry to form a hexagon among other possibilities. Two variations of an aperiodic inflation scheme are disclosed to match nine tiles into two larger tiles. These two larger tiles of the second generation are the new base shapes following the same matching rules as the original tiles. The inflation can thus be repeated indefinitely creating an arbitrarily large surface covered with a 6-fold quasiperiodic tiling consisting of only two diamond shapes. The notches of the tiles can be creatively deformed to make Escher-esque figures.