Take two squares and fourequilateral triangles and hook them together. Theytessellate, in a somewhatfamiliar pattern.
Now, fuse the six shapes into one equilateral,but concave, octagon.
Cut that shape in half, bilaterally,and use a half to form a new tile.It tessellates, of course, butnondeterministically!
http://cs.sru.edu/~ddailey/tiling/rhombsqbest.svg