The microphase-separated morphology of ABC three-arm star-shaped copolymers with arm-length ratio 1:1:x is investigated by a recently proposed simulation method, the diagonal bond method. Five kinds of two-dimensional (cylindrical) phases, three kinds of lamellar-type phases and two kinds of continuous matrix phases are discovered. The phase diagram is presented:  The progression of the morphologies as a function of x is the following:  lamella + sphere; five polygonal cylinders, [8.8.4], [6.6.6], [8.6.4; 8.6.6], [10.6.4; 10.6.4; 10.6.6], [12.6.4]; perforated layer; lamella + cylinder; columnar piled disk; lamella-in-sphere. Two remarkable features of ABC star polymer systems are found:  first, in all phases junction point monomers gather on lines where three interfaces meet, and second, the section of the cylindrical phase becomes the tessellation of even-numbered polygons. The free energy of the system is also calculated in the strong segregation limit for four kinds of simple phases, and the results are consistent with the simulations.

We present a study of photonic band structures of two-dimensional Archimedean tiling patterns. The tilings we have investigated are ͑4.8 2 ͒, ͑6 3 ͒, ͑4.6.12͒, and ͑3 2 .4.3.4͒, which have been discovered computationally and experimentally in self-assembled microphase separation of ABC star block terpolymer 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 ͑4.6.12͒ structure. Furthermore, we find that complete photonic bands readily open in the ͑3 2 .4.3.4͒ structures in the same way as in dodecagonal quasicrystals. Complex tilings open up a way to construct photonic crystals.