Posts tagged: de Bruijn
A Penrose tiling can be constructed using just two different tiles in the shape of a thick and thin rhombus:
Here, the angles are multiples of ø = π/5. The edges of these tiles are marked with two different kinds of arrows. This is done to enforce a specific matching rule that ensures that the tiling is non-periodic, which is one of the defining features of Penrose tilings. Different tiles can only be placed next to each other if the touching edges have the same type of arrow and point in the same direction.
If these matching conditions were not in place, then it would be possible to construct tilings which are periodic as shown in the following image. There are translational symmetries present in 8 different directions here:
The image at the top shows a part of a Penrose tiling obeying the matching conditions with the arrows displayed. The numbers are just there to index some other property of directionality not discussed here.
If these matching conditions are satisfied for a complete tiling, then the resulting configuration will always be non-periodic. However, following the matching rules alone does not guarantee an infinite tiling of the entire plane. It is therefore possible to construct finite regions that obey the matching rules, but cannot be extended any further without allowing for periodicities or contradicting the matching rules.
There do exist sets of tiles that will always admit non-periodic tilings in which no extra matching conditions need to be imposed. For a list of such tiles that tile the plane, 3-dimensional space, and even the hyperbolic plane, see this list of aperiodic sets of tiles.