There exist self-similar tiles in which the contraction and rotation elements of the transforms of the corresponding IFS are not related to Perron numbers, and in which the ratios of area between elements and the full attractor are (usually) not all powers of a single number. To the best of my knowledge there is little that can be said in general about the transforms involved in such tiles, so we only have the general statement that

{
T_{i}: **p** → a_{i}**(p|p̅)** + **v**_{i}
}

However we can note that the equivalent to the Perron and Pseudo-Perron tile dissection equation.

Σ|a_{i}|^{2}
= 1

Examples of non-Perron tiles include

- the order 2 right-angled triangle (involving reflections; the order 2 right isoceles triangle, which doesn't involve reflections is a Perron Tile)
- the rep-n parallelograms
- the rep-n concertina tiles
- a variety of non-parallelogram quadrilaterals.

The order-2 right-angled triangle construction
generates all possible right-angled triangles. This includes instances where
the ratios of area between elements and the full attractor are powers of a
single number, including some Perron numbers. However they differ from Perron
tiles in that the elements are inversely similar to the attractor, and the
angles aren't multiples of a base angle. The order-2 right-angled triangle can
be dissected to give an order-4 right-angled triangle, in which the elements
are directly similar to the attractor (the reflections cancel out). With the
exception of the right isoceles triangle (`4x =
1`) and the golden triangle (`x ^{2} +
2x^{3} + x^{4} =1`) the angles (0, -π/2, π/2 and 0) are
not those of a Perron tile. This observation draws attention to the question as
to whether the general rep-n

© 2016 Stewart R. Hinsley