Pseudo-Perron Tiles

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^n_i (p|p̅) + P_i(c) } or { T_i: p → c^n_i e^im_iπ (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^-1e^iθ. θ can take any value (other than nπ). For a few values of θ c is a Perron number, but for most values it is not. I make the arbitary choice that the full range of attractors constitute a single non-Perron tile and particular members for which c is a Perron number are not pseudo-Perron tiles.

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 P₀(c) and P₁(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.