We present a study of photonic band structures of two-dimensional Archimedean tiling patterns. The tilings we have investigated are (4.8.8), (6.6.6), (4.6.12), and (184.108.40.206.4), which have been discovered computationally and experimentally in self-assembled microphase separation of ABC star block terpolymers systems. Using plane wave method, we have calculated eigenvalue equations for various combinations of dielectric contrast on the complex patterns. We demonstrate the existence of complete photonic band gaps in the (4.6.12) structure. Furthermore, we find that complete photonic bands readily open in the (220.127.116.11.4) structures in the same way as in dodecagonal quasicrystals. Complex tilings open up a new way to construct photonic crystals.