An illustration for @Toscho's nice idea, from Steven Dutch's
*[Rep-Tiles](https://www.uwgb.edu/dutchs/symmetry/reptile1.htm)* webpage:
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![reptiles][1]
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"Four squares can be arranged around a central square to form a Greek Cross. Greek Crosses make a lovely plane tesselation, but they are not rep-tiles. We can try modifying a Greek Cross tesselation by replacing each cross with the compound of five crosses. It's closer, but still not an exact rep-tile. We can repeat the process, each time getting figures that are more and more crinkled and closer to a true rep-tiling. But at every step, we see that the perimeter of the tiling always has twice as many crinkles as each tile."
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  [1]: https://i.sstatic.net/xcm2a.gif