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

## Partial
Postcomposition Tiles

## IV. Partial
Postcomposition Derivatives of External Tiles

In addition to the symmetric, demisymmetric and windowed tiles
there are (at least) four other order 3 6th unit cubic tiles, including two
“external” tiles. As a 3 element tile potentially has 6 partial
postautocomposition derivatives, 3 of order 5 and 3 of order 7, these two tiles
give rise to 12 candidate attractors. However 4 of these are not connected,
leaving 8 tiles - 5 order 5 tiles and 3 order 7 tiles.

The first external tile gives rise to 2 order 5 tiles and 1 order 7 tile.
The order 5 tiles have the dissection polynomial `c+2c`^{2}+c^{3}+c^{4}.

The minimal unit cells have 2 copies of the tile. When the IFS for the tile
is `{ `**p**→-a**p** + 1;
**p**→-a**p** - 1; **p**→-a^{3}**p** - 1 + a +
3a^{2}

} the cell transforms for the first are `{ `**p**→**p**; **p**→-a**p**
- 1

} and for the second are `{
`**p**→**p**; **p**→-a**p** + 1

},
giving a signature in both cases of **01**.

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

The minimal unit cell (shown below) contains 3 copies of the tile, with a
signature of **011**. The cell transforms are `{
`**p**→**p**; **p**→-a**p** - 1

```
;
```**p**→-a**p** + 1

}.

The second external tile gives rise to 3 order 5 tiles and 2 order 7 tiles.
Two order 5 tiles have the dissection polynomial `c+2c`^{2}+c^{3}+c^{4}; the
3rd `2c+2c`^{3}+c^{5}.

The minimal unit cells have 2 copies of the tile. When the IFS for the tile
is `{ `**p**→-a**p** + 1;
**p**→-a**p** - 1; **p**→-a^{3}**p** - 1 + a +
3a^{2}

} the cell transforms for the first are `{ `**p**→**p**; **p**→-a**p**
- 1

}, for the second `{
`**p**→**p**; **p**→-a**p** + 1

}, and
for the third `{ `**p**→**p**;
**p**→-a^{3}**p** - 1 + a + 3a^{2}

}

The order 7 tiles have the dissection polynomials `4c`^{2}+c^{3}+2c^{4} and
`c+2c`^{2}+3c^{4}+c^{6}.

They both have 3 copies in the unit cell, with signatures **011** and
**013** respectively

© 2015, 2016, 2017 Stewart R. Hinsley