On non-convex polygons that tile convex polygons

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).

• "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? – Gerry Myerson Jul 18 at 13:18