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

## Complex
Teragons

While all cubic tiles are teragons (with an uncountably infinite number of
edges), of the eight known order 3 6th unit cubic
tiles the complex teragon has the most complex structure. Five of the tiles -
the symmetric (8-fold degenerate), (two) external and (two) demi-symmetric tiles are simple teragons with
non-self-approaching boundaries of relatively low dimension The windowed tile (2 dissections), while technically a
complex teragon, is composed of a spiral of diminishing copies of the symmetric
tile. In contrast the other two complex teragons have boundaries of greater
dimension which approach themselves at an infinite number of points, with
appreciable parts of the tile composed of fractally intermixed parts of 2
elements.

The first complex teragon was found with IFS { **p** →
-a^{3}**p**; **p** → a**p** + 1; **p** → a**p** + 1
+ a^{-1}

} during a heuristic directed survey. A more convenient
IFS, the attractor for which is shown below, is { **p** → a**p**;
**p** → a**p** + 1; **p** → -a^{3}**p** + 1 +
a^{1}

+ a^{2} }.

The second complex teragon was also found during a heuristic directed
survey, with IFS { **p** → -**p**; **p** → a**p** + 1;
**p** → a^{3}**p** + (1 + a) / 2^{}

}

#### Tiling

For the first there are two tilings with signature **00** (two equal
sized copies of the tile making up a unit cell). For the attractor shown above
the cell transforms are { **p** → **p**; **p** →
-**p**

} and { **p** → **p**; **p** → -**p** +
1

}. The unit cells are shown below.

The tiling vectors for the first unit cell are `2 +
a` and -a^{-1}, and for the second `2`

and `-a`^{-1}.

It is likely that there are tilings with more complicated unit cells,
containing copies of different sizes, but the complex nature of this teragon
makes it difficult to identify the correct position of copies.

The second has a tiliing with a unit cell with signature **00**, cell
transforms `{ `**p** → **p** + a +
(a^{2} - 1)/2**; ****p** → -**p** }, and tiling
vectors `(3 + a) / 2` and `(a`^{-3} - a^{-2}) / 2

© 2015, 2016, 2017 Stewart R.Hinsley