Some heuristics for generating Perron tiles have been identified. While in principle the heuristics can be applied to other Perron numbers, only quadratic, cubic and metallic tiles are known to me.

These heuristics limit the nature of the transforms that are involved in the corresponding IFS, and therefore offer an opportunity to define a compact notation to denote such tiles.

Note that all tiles have 4 degrees of freedom - scale, orientation,
x-position and y-position. The x-position and y-position degrees of freedom
mean that it is always possible to position the tile such that the vector
element of one transform is `0`. The orientation
degree of freedom means that it is alway possible to position the tile such the
the vector element of a second transform is parallel to the x-axis. The scaling
degree of freedom means that it is always possible to scale the tile so that
the vector element of that transform is `1`.

This means that that an order 2 cubic tile can be completely specified by
the combination of unit and power (of `c`) from
the transforms of the corresponding IFS. The 11 order 2 cubic tiles (12
dissections) are then [` 1^{+}5^{+}`], [

Moving on to higher order tiles, you have to specify the vector elements of
the additional transforms. This can be represented as the powers of `c` in the polynomial. For example there are two
second cubic triapodes, which are [** 1^{-}4^{-}6^{-}(0)**]
and [

The order 3 symmetric second cubic tile is [** 7^{+}2^{+}2^{+}(-0)**],
i.e. with the 3rd transform having vector element

The coefficient of a power of `c` in the
polynomial is not always a unit, so it necessary to extend the notation to
cater for this, giving **(n*k)**, **(n/l)** and **(n*k/l)**
denoting `kc ^{n}`,

The notation is not capable of covering all pseudo-Perron tiles, but those
pseudo-Perron tiles involving reflections in the x-axis can be represented by
adding an overline to the power. It may also necessary to explicitly state the
vectors, as it's not always possible to simulaneously arrange for the vector
elements of the first two transforms be `0` and
`1`, and for the reflections to be in the x-axis.
For example the golden bee is [` 1^{-}(1)2^{}^{+}(0)`].

The notation above is not sufficient for quadratic tiles as there are
multiple Perron numbers with the same magnitude. We can address this by adding
the value of the second coefficient ("m") from the Perron polynomial. For
example the √2:1 rectangle becomes [`^{0}1^{+}1^{+}`], the
tame twindragon [

The notation above is also not sufficient for pletals and hextals, in which
units other than `+1` and `-1` are involved.

For pletals the additional units are `+i` and
`-i`. These can be handled by adding
**i** to the appropriate places in the
notation. For example the fat cross is [**^{2}1^{+}3^{+}3^{+}(i0)3^{+}(-0)3^{+}(-i0)**].

For hextals the additional units are the Eisenstein integers `e _{11}`,

© 2016 Stewart R. Hinsley