There exist tiles which involve the same contractions and rotations as in Perron Tiles, but in which the corresponding IFSs include reflections as well as rotations, contractions and translations. The IFS then becomes

{
T_{i}: **p** → ±c^{ni} **(p|p̅)** +
P_{i}(c) } or { T_{i}: **p** →
c^{ni} e^{imiπ} **(p|p̅)** +
P_{i}(c) }

These figures don't share all the properties of Perron Tiles, but do tile
the plane. Two element examples include the scorpion and arachnodragon for
Perron number `1 + i`, and the golden bee and
golden triangle for Perron number `i(1 +
√5)/2`.

To distinguish these from Perron tiles, and from self-similar tiles in which Perron numbers are not involved, I introduce the term Psuedo-Perron tile.

There are some ambiguous cases. For example the value of `c` for the rep-n parallelograms is `√n ^{-1}e^{iθ}`.

The parameter space for Pseudo-Perron tiles includes not only the relative
position (the translation component of the corresponding transforms) of the
elements to each other, but also the position of the elements relative to the
axis/axes of reflection. This make the parameter space larger (e.g. the golden
bee would not be found if the translation elements to were set to `0` and `1`, as can be
done with Perron tiles) which makes running a computational survey for tiles
less practicable. (I conjecture that the equivalent survey for two element
Pseudo-Perron tiles involves reflections in the x-axis, and translations of
P_{0}(c) and P_{1}(c) instead of `0` and `1`.

Pseudo-Perron tiles are common for quadratic and quartic Perron numbers, but I have not found any for cubic Perron numbers.

© 2016 Stewart R. Hinsley