This is a follow-on from a previous question, in which I paraphrased Statement 10.1.1 of Grünbaum and Shephard's Tilings and Patterns. The original statement is shown below:
where Figures 10.1.3 to 10.1.6 are the following.
I'm having a hard time understanding what exactly the authors mean by "uniqueness of the $k$-composition process" for a $k$-similarity tiling. They say that
"For some $k$-similarity tilings, such as those of Figures 10.1.3 and 10.1.4, there are many different ways in which unions of sets of $k$ tiles of $\mathcal{T}$ can be taken as the tiles of a tiling similar to $\mathcal{T}$. In other cases, such as those of Figures 10.1.5 and 10.1.6, this composition process is unique."
But surely there's more than one way of taking a composition of the tiling in Figure 10.1.5? For example, won't a scaling of 2:1 about the point marked in red (see below) also define a 4-composition of the original tiling?
Help! I'm lost!
Figured this out a while back, posting the answer here now.
A $k$-composition of a tiling $\mathscr{T}=\{T_i\}_{i \in \mathbb{N}}$ can be thought of as a partition of the set $\mathscr{T}$, where each block of the partition contains precisely $k$ tiles. Two $k$-compositions $\mathscr{T}_1$ and $\mathscr{T}_2$ of $\mathscr{T}$ are equal precisely when they induce the same partition of $\mathscr{T}$.
The "composition process" is "unique" if all possible $k$-compositions of $\mathscr{T}$ are equal.