There is a mathematical result from Thurston and Kenyon that the mapping of
a self-similar tiling in ℝ^{2} to itself is always the multiplication
of each point of ℝ^{2} by an integral power of a Perron number and
that there is at least one self-similar tiling for every Perron number.

This suggests that there are self-similar tiles whose construction involves
Perron numbers. However it may not imply that there is a self-similar tile for
every Perron number as some of these tilings involve tiles which are the
attractors of graph directed IFSs (Rauzy tiles, *sensu lato*, as studied
by Cantorini, Siegel and Arnoux), rather than "simple" (stateless) IFSs.
However self-similar tiles are known for some Perron numbers.

As noted the transforms of any IFS generating a self-similar figure are of
the form `ae ^{iω}p +
x`. Let

{
T_{i}: **p** → u_{i}c^{ni} **p** +
P_{i}(c) }

where `u _{i}` is a unit..

The next step is to determine `n _{i}`,
u

It is obvious that the coefficients of `D(x)`
are non-negative integers, that `a _{0} =
0`, and that the sum of the coefficients is the number of transforms of the
IFS and the number of elements in the self-similar dissection of the tile. This
leads to a strategy for finding candidate values of

In the general case `u _{i}` may be

{
T_{i}: **p** → ±c^{ni} **p** +
P_{i}(c) }

I conjecture that the coefficients of `P _{i}(c)` are rational, which means that they
can be scaled so that they are all integers. (But see notes below about pletals
and hextals.)

`P _{0}(c)` and

There are 3 sets of Perron numbers for which I know that there are related tiles.

1) Those Perron numbers which are roots of quadratic equations, and which
are either complex or integer. Each such number is satisifed by, *inter
alia*, a dissection polynomial `nx = 1`,
corresponding to tiles with `n` elements all of
the same size, which are known as rep-tiles. I have a construction which
generates tiles for all such Perron numbers, but can only formally prove the
existence of a tile for Perron numbers of the form `i√n`, for which a geometrical construction is
available to produce the rep-n-rectangle. I am however confident in the
existence of "linear" and "caterpillar" rep-tiles for all such Perron
numbers.

Some rep-tiles are associated with a square grid.
In this case the allowed units are the unit Gaussian integers (`+1`, `-1`,
`+i` and `-i`), and
the co-effecients of the polynomial become Gaussian rationals, i.e. numbers of
form `a + ib`, where `a` and `b` are rational.
These form a richer set of tiles, and I coin the term pletals (polyPLET
fractALs) to refer to them. The units can also be written `e`^{imπ/2}. Thus the IFS for a pletal
is

{
T_{i}: **p** →
c^{ni} e^{imiπ/2} **p** +
P_{i}(c) }

Some rep-tiles are associated with a hexagonal
grid. In this case the allowed units are the unit Eisenstein integers ( (`+1`, `+e _{11}`,

{
T_{i}: **p**
→c^{ni} e^{imiπ/3} **p**
+P_{i}(c) }

2) Perron numbers whose magnitude is the square root of a unit cubic Pisot
number. I have found tiles for the 1^{st} (plastic number),
2^{nd}, 3^{rd} (Douady number), 4^{th} (Tribonacci
number) and 6^{th} unit cubic
Pisots, and have a construction for the dissection polynomial `nx + x ^{2} + x^{3}`, which gives
some additional such numbers, including the 8th and 12th unit cubic Pisots. (The
corresponding Pisot polynomials are

3) Perron numbers of the form `i√m`, where
`m` is a metallic number. There is a construction
for `m:1` rectangles, so there is a tile for each
such number.

© 2016 Stewart R. Hinsley