A Proposed Notation for Quadratic, Cubic and Metallic Perron Tiles

Some heuristics for generating Perron tiles have been identified. While in principle the heuristics can be applied to other Perron numbers, only quadratic, cubic and metallic tiles are known to me.

These heuristics limit the nature of the transforms that are involved in the corresponding IFS, and therefore offer an opportunity to define a compact notation to denote such tiles.

cubic tiles

Note that all tiles have 4 degrees of freedom - scale, orientation, x-position and y-position. The x-position and y-position degrees of freedom mean that it is always possible to position the tile such that the vector element of one transform is 0. The orientation degree of freedom means that it is alway possible to position the tile such the the vector element of a second transform is parallel to the x-axis. The scaling degree of freedom means that it is always possible to scale the tile so that the vector element of that transform is 1.

This means that that an order 2 cubic tile can be completely specified by the combination of unit and power (of c) from the transforms of the corresponding IFS. The 11 order 2 cubic tiles (12 dissections) are then [1+5+], [1+5-], [1-5+], [1-5-], [2+3+], [2+3-], [2-3+], [2-3-], [1+3+], [1+3-], [1-3+] and [1-3-]. (The contraction and rotation elements of the transforms is implied by the combination of powers.)

Moving on to higher order tiles, you have to specify the vector elements of the additional transforms. This can be represented as the powers of c in the polynomial. For example there are two second cubic triapodes, which are [1-4-6-(0)] and [1-4-6+(0)], where (0) denotes c0 (equals 1). There is a potential ambiguity with (-n), which could be mean -cn or c-n. However the degrees of freedom described above mean that it is alway possibly to position orient and scale the tile so that the largest vector is 1, so it is not strictly necessary to deal with negative powers, and this doesn't need to be catered for by the notation. If you do want to refer to a negative power add an extra pair of brackets, i.e. ((-n)).

The order 3 symmetric second cubic tile is [7+2+2+(-0)], i.e. with the 3rd transform having vector element -1. Its order 3 demi tiles are [7+2+2-(0)] and [7-2+2-(0)]. Another order 3 second cubic tile is [7-2+2+(0-5)], where the 3rd transform has vector element 1 - c5.

The coefficient of a power of c in the polynomial is not always a unit, so it necessary to extend the notation to cater for this, giving (n*k), (n/l) and (n*k/l) denoting kcn, cn/l and (k/l)cn respectively.

pseudo-Perron tiles

The notation is not capable of covering all pseudo-Perron tiles, but those pseudo-Perron tiles involving reflections in the x-axis can be represented by adding an overline to the power. It may also necessary to explicitly state the vectors, as it's not always possible to simulaneously arrange for the vector elements of the first two transforms be 0 and 1, and for the reflections to be in the x-axis. For example the golden bee is [1-(1)2+(0)].

quadratic tiles

The notation above is not sufficient for quadratic tiles as there are multiple Perron numbers with the same magnitude. We can address this by adding the value of the second coefficient ("m") from the Perron polynomial. For example the √2:1 rectangle becomes [01+1+], the tame twindragon [11+1+] and the twindragon [21+1+]. The pseudoterdragon is [11+2+2+(-0)]. The √3:1 rectangle is either [01+1+1+(-0)] or [01+1+1+(0*2)] (alternatively [01+1+1+(0+0)]).

pletals and hextals

The notation above is also not sufficient for pletals and hextals, in which units other than +1 and -1 are involved.

For pletals the additional units are +i and -i. These can be handled by adding i to the appropriate places in the notation. For example the fat cross is [21+3+3+(i0)3+(-0)3+(-i0)].

For hextals the additional units are the Eisenstein integers e11, e01, -e11 and -e01. As -e01 is e112 we can express all the additional units in terms of e01, which we can denote e. Then, for example, the terdragon becomes [3-e2+1+1+(-0)].

© 2016 Stewart R. Hinsley