I think I got confused with the definition of aperiodic tiling. Look at the following example:
First, try to find a "1-dimensional aperiodic tiling". Start with the string 0, then make the string 01, then 0110, then 01101001, i.e. each time double the size of the previous string by adding a "complement" string; in this way we get the Thue-Morse sequence 0110100110010110...; call this $\{a_n\}$. Out of $a_n$, by translations we can make a sequence with index in ${\mathbb Z}$ rather than ${\mathbb N}$: Let $b^n_i=a_{2^n+i}$, thus the index $i$ runs from $-2^n$ to $\infty$; let $n\to\infty$, there is a subsequence $n_j$ so that $\lim_{j\to\infty}b^{n_j}_i=c_i$ for all $i\in {\mathbb Z}$. One can check that any pointed limit (the same way one takes pointed Gromov-Hausdorff limit) in this $c_i$ sequence is not periodic, so $c_i$ should be "aperiodic".
Then use this to make a 2-dimensional tiling: first tile the plane by squares. Take the square at position $(i, j)$; if $c_j=0$, keep this square as it is; if $c_j=1$ and $c_i=0$, cut the square into two rectangles by its vertical bisector; if $c_j=1$ and $c_i=1$, cut the square into two rectangles by its horizontal bisector; thus we get a tiling of plane by squares and $1\times \frac 12$ rectangles. This looks "aperiodic" to me, can anyone tell me where I got wrong? Thanks!
There are several non-equivalent definitions in the literature. Some just ask for non-periodicty in 1 or more directions (no full-rank lattice of periods). Some ask for non-periodicity in all directions (no period at all). And some ask for the stronger condition that there is no sequence of increasing subpatches of the tiling that converges to a periodic tiling; that is, every tiling whose patches all appear in the original must be non-periodic.
This last definition is the most common.