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Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such 'rectifiable' polyominoes, to form the smallest rectangle copies of the polyomino are arranged in a very non-trivial layout.

A natural generalization would be to non-convex and non-rectilinear polygons (those totally non-rectilinear polygons with none of the angles equal to 90 or 270 or partially rectilinear ones).

Question: Is there any such non-convex and non-rectilinear polygon $P$ such that the convex polygon tiled with least number of copies of $P$ has a non-trivial layout?

Note: An example of 'trivial layout': Consider a regular $n$-gon partitioned into n identical isosceles triangles with apexes at its center. Replace all arms of all triangles with identical zig-zags. The regular polygon is tiled by copies of a complex non-convex polygon but the layout topology is of a simple ring. As mentioned above, for many rectifiable polyominos, the topology of the layout when a rectangle is tiled by their copies is quite complex (for example see here: http://polyominoes.org/rectifiable).

Further question added after answer below: Are there convex hexagons that can nontrivially tile some convex polygon?

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  • $\begingroup$ "Recall that for many rectifiable polyominos, the topology of the rectangles tiled by their copies is quite complex." How can I recall something I've never known? Can you give or cite an example or two? $\endgroup$ Commented Jul 18, 2019 at 13:18

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In Erich Friedman's Math Magic Puzzle column ( https://erich-friedman.github.io/mathmagic/0499.html ) a very similar question was asked. I suggest scrolling down to the bottom of the answers page where you'll find several non-convex examples. Like this one:

non-convex tiles with various spectra

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  • $\begingroup$ Thanks for this pointer. This basically has answered the question, as far as I can make out. $\endgroup$ Commented Apr 26 at 9:54
  • $\begingroup$ And there is also info on convex polygons with 4 and 5 sides that can non trivially tile convex polygons with much larger numbers of sides - a bonus. But are there convex hexagons that can produce nontrivial tilings of convex polygons? $\endgroup$ Commented Apr 27 at 9:43

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