Organic Tessellation

Hello there!

I have run into a design challenge lately and I was hoping someone here can help me solve it. I am working on a installation pattern that is based on an organic tile (looks a bit like a fidget) inspired by the work of M.C Escher and his hexagonal tessellation.

211113_Organic tessellation.3dm (286.5 KB)
Organic (11.9 KB)

However, as you can see above I have failed in my attempts to tessellate the initial tile or achieve a non overlapping pattern. I am not limited to one initial curve but I prefer to achieve the results with no more that 3. The aim at the end is to 3D print each piece separately so would be best to have minimum initial shapes.

I have made some attempts using the Parakeet Escher components but I am not getting anywhere. or maybe there is another logic I am missing here?