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I'm reading Grünbaum and Shephard's Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). Paraphrasing:

If a monohedral tiling $\mathcal{T}$ is a $k$-composition of itself in a unique way then $\mathcal{T}$ must be non-periodic, i.e. does not have translational symmetry. (A tiling $\mathcal{T}_1$ is a $k$-composition of $\mathcal{T}_2$ if every tile in $\mathcal{T}_1$ is a union of $k$ tiles in $\mathcal{T}_2$, and $k>1$ is the smallest integer for which this holds.)

In the brief proof they give is the statement "Suppose there were such a translational symmetry $t$. Then uniqueness of composition implies that $t$ must also be a symmetry of the $k$-composed tiling." They then go on to use this to derive a contradiction.

But I don't understand what uniqueness of composition has to do with it, if $t$ is a translational symmetry of $\mathcal{T}$ then isn't it automatically also a symmetry of the composed tiling?

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Uniqueness is extremely important. As a counter example consider the tiling of the real line by unit intervals. This tiling is a $2$-composition of itself in exactly two different ways (either every even tile is glued to the tile on its right, or to the tile on its left). Translation to the left by $1$ is a symmetry of this tiling, but it is not a symmetry of either of its $2$-composed tilings because it permutes these two tilings.

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    $\begingroup$ Thanks for the enlightening example, but I'm still confused about what exactly "uniqueness" means for compositions. What distinguishes one composition from another? $\endgroup$ – Josh Chen Aug 25 '14 at 11:21

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