Three-dimensional photonic crystals, or periodic materials, that do not allow the propagation of photons in all directions with a wavelength in the visible region have not been experimentally fabricated, despite there being several potential structures and the interesting applications and physics that this would lead to. We show using computer simulations that two structures that would enable a bandgap in the visible region, diamond and pyrochlore, can be self-assembled in one crystal structure from a binary colloidal dispersion. In our approach, these two structures are obtained as the large (Mg) and small (Cu) sphere components of the colloidal analogue of the MgCu(2) Laves phase, whose growth can be selected and directed using appropriate wall patterning. The method requires that the particles consist of different materials, so that one of them can be removed selectively after drying (for example, by burning or dissolution). Photonic calculations show that gaps appear at relatively low frequencies indicating that they are robust and open for modest contrast, enabling fabrication from more materials.