  Colloidal self-assembly allows rational design of structures on the
micrometer and submicrometer scale. One architecture that can generate complete
3D photonic band gaps is the diamond cubic lattice, which has remained
difficult to realize at length scales comparable to the wavelength of visible
or ultraviolet light. Here, we demonstrate three-dimensional photonic crystals
self-assembled from DNA origami that act as precisely programmable patchy
colloids. Our DNA-based nanoscale tetrapods crystallize into a rod-connected
diamond cubic lattice with a periodicity of 170 nm. This structure serves as a
scaffold for atomic layer deposition of high refractive index materials such as
TiO$_2$, yielding a tunable photonic band gap in the near-ultraviolet.
