Polygons such that $n^2 $ times magnification of a polygon could be covered by exactly $n^2$ original polygon

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While studying about covering problems in combinatorics, I got to a simple question:

What polygons can be covered exactly, without any area that is covered twice or area that is outside the covered area?

For example, all parallelogram that is $n^2$ times magnified can be exactly covered by original parallelogram, as following.

Are there any research known about this property, such that the polygon can be exactly covered by original polygon when magnified $n^2$ times?

user64494

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asked Mar 20, 2019 at 6:27

SSHS_Space

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$\begingroup$ I think this only works for triangles and some quadrilaterals. The moment you have a polygon with more than $4$ sides you run into problems with the angles. $\endgroup$
– quarague
Commented Mar 20, 2019 at 10:15
2

$\begingroup$ Stick three equal squares together in an L shape. You can fit four of those in an L magnified by 2, nine of them into an L magnified by 3, and, for all I know, for every $n$, $n^2$ of them into an L magnified by $n$. $\endgroup$
– Gerry Myerson
Commented Mar 20, 2019 at 11:28
$\begingroup$ @GerryMyerson Thanks for the example. I was trying to draw something like that and wasn't able to do so, so I wrongly concluded that it is impossible in general. $\endgroup$
– quarague
Commented Mar 20, 2019 at 11:57
2

$\begingroup$ Seems related: en.m.wikipedia.org/wiki/Rep-tile although here it is only asked for one and not all $n$. $\endgroup$
– user35593
Commented Mar 20, 2019 at 15:29

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