**Some tiles associated with the 6th unit cubic Pisot number**

## Partial
Postcomposition Tiles

## III. Metawindowed
Tiles

A 3 element tile potentially has 6 partial postautocomposition derivatives,
3 of order 5 and 3 of order 7. However the windowed
tile has 2 dissections, which increases the number of candidates to 12.
However 6 of these are not connected, leaving 6 tiles - 3 order 5 tiles and 3
order 7 tiles.

There are 2 order 5 tiles with dissection polynomial `2c+2c`^{4}+c^{6} ...

... and one with dissection polynomial `c+2c`^{2}+c^{3}+c^{4}.

The minimal unit cells (shown below) contain 4 copies of the tile. As there
are unit cells for the windowed tile with signatures **00**, **01** and
**02** there are alternative signatures for the unit cell. Getting back to
the windowed file from the first two has the partial signature **03**, so
the signature can be **0033**, **0134** or **0235**. Getting back to
the windowed tile from the third has the partial signature **01**, so the
signature can be **0011**, **0112** or **0123**. Unit cells with
signatures **0235** and **0123** are shown below.

There are 2 order 7 tiles with dissection polynomial `4c`^{2}+c^{3}+2c^{4}.

The third order 7 tile has the dissection polynomial `c+2c`^{2}+3c^{4}+c^{6}.

The minimal unit cells (shown below) contain 6 copies of the tile. The first
two have signature **011233** and the last signature **012335**.

© 2015, 2016, 2017 Stewart R. Hinsley