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Some geometrical facts about the perfect Penrose tiling

 

The Penrose rhombus tiling shown has many interesting geometrical properties, of which a few are listed here.

 

A sample of the undisordered Penrose rhombus tiling

 

Fourier transform : The structure factor consists of Dirac peaks . The main peaks nearest to the origin are often used to define a quasi-Brillouin zone, but there are an infinity of lesser peaks distributed densely in the reciprocal plane, with five-fold symmetry.

 

Repetitivity : Any given local environment of linear size R is guaranteed to repeat a little further on in the tiling within a distance of about 2R. The repetition is not, of course, periodic as in a crystal. Concerning nearest neighbor environments, a spin can have three, four, five, six or seven neighbors, with a frequency of occurrence of each type of local environment that can be exactly determined in terms of the golden mean, τ = (√5 – 1)/2

 

Inflation : One can transform the tiling into a scaled up or scaled down version of itself by eliminating sites or introducing sites according to well-defined rules. The scale factor is the golden mean τ.

 

Mapping into internal space : Each site has a counterpart in the internal – or perpendicular – space representation of the tiling, in which sites map into four plane pentagonal regions, depending on their spatial characteristics. The two sublattices map into different planes. Planes 1 and 3 for sublattice A and planes 2 and 4 correspond to sublattice B, which is equivalent to A upto a rotation. Sites of different coordination number z map into different domains, as shown in the figure below which uses a different color for each z. Each of the colored domains can in turn can be divided into smaller subdomains depending on the next neighbors, and so on . The invariance of the structure under inflation-deflation leads to a self-similarity of the tiling in the internal space representation.

 

Sites of the tiling as seen in the physical plane…

 

Tiling with spins colored according to number of neighbors (yellow=3,blue=4,red=5, green=6,violet=7)

 

…and as seen in « perpendicular » space (the figure is schematic. It represents parallel planes at four different heights along the vertical axis of this 3D space)