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+2c4+c6 ...

order 5 metawindowed tileorder 5 metawindowed tile

... and one with dissection polynomial c+2c2+c3+c4.

order 5 metawindowed tile

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.

order 5 metawindowed tileorder 5 metawindowed tileorder 5 metawindowed tile

There are 2 order 7 tiles with dissection polynomial 4c2+c3+2c4.

order 7 metawindowed tile order 7 metawindowed tile

The third order 7 tile has the dissection polynomial c+2c2+3c4+c6.

order 7 metawindowed tile

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

order 7 metawindowed tileorder 7 metawindowed tileorder 7 metawindowed tile

© 2015, 2016, 2017 Stewart R. Hinsley