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We catalog known optical moir\'e lattices and uncover exotic latticeconfigurations following a geometric analog of the ancient sieve ofEratosthenes algorithm for finding prime numbers. Rich dynamics ofBose-Einstein condensates loaded into these optical lattices is revealed fromnumerical simulations of time-of-flight interference patterns. What sets thismethod apart is the ability to tune the periodicity of the optical latticeswithout changing the wavelength of the laser, yet maintaining the localpotential at the individual lattices sites. In addition, we discuss the abilityto spatially translate the optical lattice through applying a structured phaseonly.

Sieve of Eratosthenes for Bose-Einstein Condensates in Optical Moir\'e Lattices