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Aperiodic Tilings Within Conventional Lattices

A later page begins a series on using either matching rules or recursive tiling to achieve tilings with aggregate fivefold or eightfold symmetries, symmetries that can only be fully achieved around a single point at best, but which can be achieved in a partial sense around an infinite set of points in a tiling if that tiling is a suitably-arranged aperiodic tiling.

Recursive tiling need not generate novel symmetries. Here are a couple of recursive tilings that operate within the lines set down by a conventional square or triangular tiling:

Recently, an aperiodic tiling involving two tiles, and matching rules for those tiles, was derived from the first of these two simple recursive tilings by Chaim Goodman-Strauss. The two tiles used are called the trilobite and the cross, and the matching rule involves not colored areas that touch along a line, but rather those that are on opposite sides where four of the sharp points of the pieces meet.

The image above is an attempt to make the boundaries of these tiles clearly visible despite the way in which the colors of the tiles tend to obscure those boundaries; also, it has been intended to make the relationship to the previously-known L-tiling apparent.

An alternate set of tiles that can produce the same tiles with conventional color matching rules is shown below:

The first tile replaces the trilobite, but the second tile replaces not only the crosses, but also the places where the sharp corners of four tiles meet. The proof that this tiling is equivalent to the trilobite and cross tiling was quite difficult.

The diagram below shows two sets of six tiles whose matching rules lead to aperiodic tilings. The top row shows the six tiles for the Robinson tiling, the bottom row shows the six tiles for the square Ammann tiling. These tiles can be reflected and rotated; for the Ammann tiling, only one piece needs to be able to undergo 90 degree rotations, the last one.

Here is the Robinson tiling which achieves an impressive aperodic structure:

This tiling was described in a paper published in 1971, one year before the publication of the Penrose tilings, and this was the first tiling known which forced aperiodicity with only six tiles.

And this is the square tiling due to Robert Ammann, who also devised the Ammann-Beenker tiling with octagonal symmetry, as well as several other tilings of different kinds:

Another tiling devised by Robert Ammann is this one, in which the ratios between sides of different length of the pieces are the golden ratio or its powers:

An attempt has been made to show a large extent of this tiling so that a subtle characteristic of it might be visible: the density of smaller tiles, shown in yellow in this diagram, varies along directions characteristic of pentagonal symmetry. This is an example of a phenomenon known as Ammann bars, which will be discussed in the case of the Penrose tiling later on.

Note that despite its reflection symmetry about a diagonal, the smaller tile is reflected instead of rotated by 90 degrees for purposes of recurrence. This is not merely laziness on my part in the use of the paint program with which I have drawn these images, it is actually how the recurrence relations work for this tiling.

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