After reading an interesting article on pentagon tilings by David Bailey, it reminded me of a shape I had constructed some years ago, from four isosceles triangles with sides 1,2 and 2. Jaap Scherphuis commented that the tile is not only of type 1 (from A+B=180) but also of type 2 (not only from B+D=180 with b=d, but also from B+D=180 with c=e).
B+A=180, B+D=180, B+E=180, C=2B, a=b=d, c=e, a=2c
Below are a few periodic tilings that be constructed with this shape. It can also tile chaotically although I haven’t got too far with it yet. Coloured areas denote mirroring.