There is a tile with the same outline as the symmetric tile, but with the smaller element embedded
in one of the larger elements, leaving a spiral of voids ("windows") in the
overall tile. Alternatively it can be viewed as a spiral of symmetric tiles.
There are two alternative dissections. An IFS for the 1st dissection is {
`p → -ap - 1; p → -ap + 1; p → ap`

} and for the 2nd is { ^{3} + 1 -
a^{3}```
p → -ap - 1; p → ap + 1;
p → -ap
```

}.^{3} + 1 + a^{3}

As for the symmetric tile it is useful to introduce the numbers
`t`

to describe tilings._{n} = a^{n}(3 - a - a^{2}) = 4a^{n}/(1
- a^{3})

It is intuitively obvious that it is not possible to tile the plane with
only one copy of the tile in the unit cell. Three tilings with two copies in
the unit cell, with signatures **00**, **01** and **02**, have been
identified.

For the IFSs given above, the cell transforms for the unit cell of the
**00** tiling are { `p → p; p → -p + 1 - (t`

} and the tiling vectors are _{0} -
t_{2}) / 2`t`

and _{0} -
t_{2}`t`

(or
_{1} + t_{2}`t`

and _{0} - t_{2}`t`

)._{1} +
t_{2}

[For the alternative IFS { ```
p → -ap; p → -ap + 1; p →
ap
```

} the cell transforms take the simpler form { ^{3} + 1```
p
→ p; p → -p
```

} and the tiling vectors are reduced in length by a
factor of 2.]

The cell transforms for the unit cell of the **01** tiling are { ```
p
→ p; p → ap + 1 + t
```

} and the tiling vectors are
_{2}`t`

and _{0}`t`

._{1} + t_{2}

[For the alternative IFS the cell transforms take the simpler form { ```
p
→ p; p → ap
```

} and the tiling vectors are again reduced in length by
a factor of 2.]

The cell transforms for the unit cell of the **02** tiling are { ```
p
→ p; p → -a
```

} and the tiling vectors
are ^{2}p + a^{2}`t`

and _{0}`t`

._{1}

[For the alternative IFS the cell transforms take the simpler form { ```
p
→ p; p → -ap
```

} and the tiling vectors are again reduced
in length by a factor of 2.]^{2}

Additional tilings can be produced by dissecting one or both elements of the
unit cell. Thus from **00** we have tilings with signatures **0113** and
**000022**; from **01** tilings with signatures **0002**, **0224**
and **002113**; and from **02** tilings with signatures **0012**,
**0335** and **002224**. Obviously further dissections can be
performed.

The unit cell for the **00** tiling is the same as one of the unit cells
for the **012** tiling of the symmetric tile, the unit cell for the
**01** tiling is the same as the unit cell for the **02** tiling of the
symmetric tile, and the unit cell for the **02** tiling is the same as the
unit cell for the **0** tiling of the symmetric tile. Consequently any
tiling of the windowed tile can be converted into a tiling of the symmetric
tile, and vice versa.

As there are two different dissections of the tile each dissection step
leads to two different tilings. For example the two unit cells with signature
**0335** are as shown below.

Note that the **02** unit cell can replace a symmetric tile in two
different orientations, with different selections of orientation give different
tilings. Thus for the symmetric tile **0** tiling there are not only
**02** tilings, but also three different **0202** tilings, with the
orientation alternating along rows, along columns, or diagonally. Combining
choice of dissection and orientation gives 8 **0123** tilings based on the
symmetric tile 01 tiling.

Pulling apart rows and/or columns and filling in the gaps is an alternative
to convering tilings of the symmetric cell. Some sample "pull-apart" tilings
(**0002**, **0011**, **001111**, **0012** and 2 versions of
**001212**) are shown below.

© 2015, 2016 Stewart R. Hinsley