For tiles with dissection polynomial

`nx + x ^{2} + x^{3}`

the corresponding Pisot polynomial is

`x ^{3} -
nx^{2} - x - 1`

and the corresponding Perron polynomial is

`x ^{3} + x^{2} + nx - 1`

The case `n = 0` corresponds to the plastic
number, and the case `n = 1` corresponds to the
Tribonacci number; a considerable number of tiles are known for both of these.
However there is a construction which generates a tile for all `n`. This construction was first reverse engineered
from a tile with `n = 2` used as an example in
Kenyon's "The Construction of Self Similar Tilings". I had previously
conjectured the existence of 4-element tiles in which the ratio of the areas of
the elements was a power of the 8th unit cubic Pisot number, that is the real
root of `x ^{3}-2x^{2}-x-1`, but had not discovered a
construction for any of these. Although not described in terms of Pisot numbers
Kenyon's construction provided convincing evidence of the existence of such
tiles. Having confirmation of the existence of such tile I investigated
further, and found a construction for Kenyon's tile.

Let `a` be the number associated with this
tiling, as described on the overview of Perron
number tilings. Then the IFS is `{ p → ap; p
→ap + x; p → a^{2}p + y;
p → a^{3}p + z }`. We can arbitrarily fix

`{ p → ap; p
→ap + 1; p →a^{2}p - a -
a^{2}; p → a^{3}p - a + a^{2}
}`

It turns out that this construction can be generalised for other values of
`n`. The IFS becomes

`{ p → ap; p
→ap + 1; ... ; p →ap + n - 1; p
→a^{2}p - a - a^{2}; p →
a^{3}p - a + (n-1)a^{2} }`

`n = 0` generates the Ammonite tile (a plastic
tile) and `n = 1` generates the Rauzy fractal.

The attractors for `n = 3 ... 10` are shown
below.

All of these figures tile the plane with one copy per unit cell. The tiling
vectors are `a ^{-1}` and

Additional tiles can be generated by modification of the corresponding IFSs;
I document some elsewhere for the 8^{th} (`n
= 2`) and 12^{th} (`n = 3`) unit cubic Pisots.

**Source:** Reconstruction (n=2) of a tile published by Richard Kenyon;
for n=3 and above these tiles are my own discoveries, from 2002. Tilings added
in 2016.

**References**

- Richard Kenyon, The Construction of Self Similar Tilings

© 2002, 2005, 2016 Stewart R. Hinsley