On Convex Layouts from Non-Convex Polygons

It is well known (for example Michael Reid's page ) that among non-convex polyominoes are several whose copies can team up to form a neat rectangle. And for several such rectifiable polyominoes (esp those with high order), the smallest rectangle that its copies form has a very non-trivial layout.

Polyominoes obviously are rectilinear polygons - all angles are 90 or 270 degrees). Consider non-convex polygons none of whose angles are 90 or 270. Is there any such totally non-rectilinear polygon P for which the convex polygon formed with least number of copies of P has a non-trivial layout?

Note: Obviously, there are non-convex polygons (equally, polyominoes) such that any number of copies cannot form a convex polygon (rectangle) together.

An example of a trivial layout: Consider a regular n-gon partitioned into n identical isosceles triangles radiating from its center. Replace all arms of the triangles with identical zig-zags. The regular polygon is now a layout of complex non-convex polygons but its topology is of a simple ring. Note that in many of the polyominos with high order, the topology of rectangular layouts with their copies is quite complex.

One can also ask for non-convex polygons with not all angles 90 or 270 for which the smallest convex region its copies form emerges from a non-trivial layout.