Ammann–Beenker tiling

[3][4] In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry.

[5] Amman's A and B tiles in his pair A5 are a 45-135-degree silver rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings.

Katz[6] has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match.

The orientation of the vertex arrows which force aperiodicity, then, can only be deduced from the entire infinite tiling.

Note that the bars for the asymmetric tile extend partly outside it.

We can then obtain an Ammann–Beenker tiling by projecting a slab of hypercubes along either the first two or the last two of the new coordinates.

A related high dimensional embedding into the tesseractic honeycomb is the Klotz construction, as detailed in its application here in the Baake and Joseph paper.

[9] The octagonal acceptance domain thus can be further dissected into parts, each of which then give rise for exactly one vertex configuration.

Moreover, the relative area of either of these regions equates to the frequency of the corresponding vertex configuration within the infinite tiling.

A portion of tiling by Ammann's aperiodic A5 set of tiles, decorated with finite, local matching rules which force infinite, global structure, that of Amman–Beenker tiling.
Amman's A and B pair of A5 tiles, decorated with matching rules; any tiling by these tilings is necessarily non-periodic, and the tiles are therefore aperiodic.
Ammann A5 substitution rules, used to prove that the A5 tiles can only form non-periodic hierarchical tilings and thus are aperiodic tiles.
This tiling exists in a 2D orthogonal projection of a 4D 8-8 duoprism constructed from 16 octagonal prisms .