[math-fun] cutting square into pieces to make equilateral triangle

I've added three (conjectured) sequences to the OEIS dealing with dissections - see A110000, A110312, A110356 I'm not even sure about the first term of A110000: is it known that there is no way to cut an equilateral triangle into 3 polygonal pieces which can be rearranged to form a square? (Turning over is allowed; 4 pieces is classical - see the entry for references, pictures, etc.) NJAS