  The work intends to extend the moir\'e physics to three dimensions.
Three-dimensional moir\'e patterns can be realized in ultracold atomic gases by
coupling two spin states in spin-dependent optical lattices with a relative
twist, a structure currently unachievable in solid-state materials. We give the
commensurate conditions under which the three-dimensional moir\'e pattern
features a periodic structure, termed a three-dimensional moir\'e crystal. We
emphasize a key distinction of three-dimensional moir\'e physics: in three
dimensions, the twist operation generically does not commute with the
rotational symmetry of the original lattice, unlike in two dimensions, where
these two always commute. Consequently, the moir\'e crystal can exhibit a
crystalline structure that differs from the original underlying lattice. We
demonstrate that twisting a simple cubic lattice can generate various crystal
structures. This capability of altering crystal structures by twisting offers a
broad range of tunability for three-dimensional band structures.
