We present a set of two-dimensional aperiodic structures with a large complete photonic band gap (PBG), which are named two-pattern photonic crystals. By superposing two substructures without regard to registration, we designed six new aperiodic PBG structures having a complete PBG larger than 15% for ɛ2/ɛ1=11.4. The rod-honeycomb two-pattern photonic crystal provides the largest complete PBG to date. An aperiodic structure becomes the champion structure with the largest PBG. Surprisingly, the TM and TE gaps of a two-pattern photonic crystal are much less interdependent than the PBGs of conventional photonic crystals proposed before, affording interesting capabilities for us to tune the TM and TE PBGs separately. By altering the respective substructures, optical devices for different polarizations (TE, TM, or both) can readily be designed.