  We develop a theory to compute and interpret the photonic band structure of a
periodic array of metallic helices for the first time. Interesting features of
band structure include the ingenuous longitudinal and circularly polarized
eigenmodes, the wide polarization gap [Science 325, 1513 (2009)], and the
helical symmetry guarantees the existence of negative group velocity bands at
both sides of the polarization gap and band crossings pinned at the zone
boundary with fixed frequencies. A direct proof of negative refraction via a
chiral route [Science 306, 1353 (2004)] is achieved for the first time by
measuring Gooshanchen shift through a slab of three dimensional bona fide helix
metamaterial.
