T he promise of photonic crystals and their potential applications(1,2) has attracted considerable attention towards the establishment of periodic dielectric structures that in addition to possessing robust complete bandgaps, can be easily fabricated with current techniques. A number of theoretical structures have been proposed(3-15). To date, the best complete photonic bandgap structure is that of diamond networks having Fd3m symmetry (2-3 gap). The only other known complete bandgap in a face-centred- cubic (f.c.c.) lattice structure is that of air spheres in a dielectric matrix (8-9 gap; the so called 'inverseopal' structure). Importantly, there is no systematic approach to discovering champion photonic crystal structures. Here we propose a level-set approach based on crystallography to systematically examine for photonic bandgap structures and illustrate this approach by applying it to the 11 f.c.c. groups. This approach gives us an insight into the effects of symmetry and connectivity. We classify the F-space groups into four fundamental geometries on the basis of the connectivity of high-symmetry Wyckoff sites. Three of the fundamental geometries studied display complete bandgaps including two: the F-RD structure with Fm (3) over barm symmetry and a group 216 structure with F (4) over bar 3m symmetry that have not been reported previously. By using this systematic approach we were able to open gaps between the 2-3,5-6 and 8-9 bands in the f.c.c. systems.