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We develop a class of supercell photonic crystals supporting complete photonic band gaps based on breaking spatial symmetries of the underlying primitive photonic crystal. One member of this class based on a two-dimensional honeycomb structure supports a complete band gap for an index-contrast ratio as low as , the lowest index contrast known to support a complete band gap for a two-dimensional (2D) photonic crystal. This same design principle is used to develop the first photonic crystal slab that supports a band gap for dual-polarization visible light. The complete band gaps found in such supercell photonic crystals do not necessarily monotonically increase as the index contrast in the system is increased.