Non-Perron Tiles

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

{ Ti: p → ai(p|p̅) + vi }

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

Σ|ai|2 = 1

Examples of non-Perron tiles include

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 (x2 + 2x3 + x4 =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-n2 parallelograms should be considered Perron tiles (the sizes and angles are correct, but the vector elements of the transforms arguably are not). Similar questions arise for the rep-9 trans-fudgeflake, and perhaps for other tiles with more that the basic 4 degrees of freedom.

© 2016 Stewart R. Hinsley