cutting square into pieces to make equilateral triangle

[N. J. A. Sloane]

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