  The topology, the symmetry involving the shape of dielectric cylinders, and
the lattice structure are among the most important ingredients in the
architecture of photonic crystals. In this paper, we present a systematic
derivation of the formulas which are needed in computing the photonic band
structures of many commonly used two dimensional lattice structures and
dielectric cylinders with various kinds of symmetries and rotations. Further
the results are applied to arrays of hollow cross-shaped cylinders embedded in
the alumina ceramic background. A large complete photonic band gap is found in
the high frequency regime.
