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

*External*
Tiles

Of the eight order 3 6th cubic tiles, two (the symmetric and windowed
tiles) have the smaller element lying within the larger two elements, two (the
complex teragons) have it lying partially within and
partially outside, and four have it lying outside. Two of these are demi-symmetric tiles, which makes it convenient to refer
to the remaining two as *external* tiles. The IFS for the first attractor
shown below is { **p**→-a**p**; **p**→-a**p** + 1;
**p**→-a^{3}**p** + 2a^{2}

}. Another IFS, with
an attractor with a different orientation, is {
**p**→-a^{3}**p**; **p**→-a**p** + 1;
**p**→-a**p** + 1 + a^{2}

}. The IFS for the second
attractor is { **p**→a**p**; **p**→a**p** + 1;
**p**→a^{3}**p** + 1 - a^{-1} - a^{2}

}.

#### Tiling

For the first tile there is a tiling with one copy in the unit cell
(signature **0**). There are also tilings with other signatures (e.g.
**03**, **033**, **0333**). For the tiling with signature **0** the
tiling vectors are `2 + a`

and `a`^{-1}

. For the
tiling with signature **03** the cell transforms are {
**p**→**p**; **p**→a^{3}**p** + 1 - a^{1}
- a^{2} - a^{3}

} and the tiling vectors are ```
2 +
2a + a
```^{-1}

and `a`^{-1}

. For the tiling with
signature **033** the cell transforms are { **p**→**p**;
**p**→a^{3}**p** + 1 - a^{1} - a^{2} -
a^{3}; **p**→a^{3}**p** + 1 - a^{2} -
a^{3}

} and the tiling vectors are ```
2 + 3a +
2a
```^{-1}

and `a`^{-1}

. For the tiling with
signature **0333** the cell transforms are { **p**→**p**;
**p**→a^{3}**p** + 1 - a^{1} - a^{2} -
a^{3}; **p**→a^{3}**p** + 1 - a^{2} -
a^{3}; **p**→a^{3}**p** + 1 - 2a^{2} -
a^{3}

} and the tiling vectors are ```
2 + 4a +
2a
```^{-1}

and `a`^{-1}

. Obviously this series
can be extended for all signatures **03**^{n}. Other simple tilings
have signatures **002** (dissecting each unit cell of **0**), **0113**
(dissecting alternative unit cells along rows, or columns, or diagonals),
**0022** (dissecting the larger copy in **03**) and **0446**
(dissecting the smaller copy in **03**).

For the second tile there is again a tiling with signature **0**. The
tiling vectors are `2 + a`

and `a`^{-1}

. There
are no obvious "pull apart" tilings, but tilings with signatures like
**002** and **0113** can be obtained by dissection.

© 2015, 2016 Stewart R. Hinsley