We describe a technique for exposing patterns of spatial period p/n using near-field diffraction from masks of spatial period p. The technique, which we propose to call ’’spatial-period-division’’, can be used with visible, UV, or x-ray radiation. With soft x rays, it is possible to produce gratings of much finer spatial period than can be achieved by current laser holographic methods. A simple model predicts that a mask pattern of period p, with an opening or slit of width ?p/2n, will give rise to an intensity pattern of period p/n at a distance from the mask, S=p2/nλ, where λ is the wavelength of the incident radiation and l≪p. The model was confirmed at visible wavelengths using a scaled simulation of a laboratory x-ray lithography unit. The feasibility of the technique was demonstrated at the 4.5 nm carbon K x-ray wavelength by ’’doubling’’ a 196.8 nm period grating-pattern x-ray mask to produce a 98.4 nm period pattern in PMMA. Exposure of higher spatial-frequency-multiples appears feasible, especially with a synchrotron source. If the mask introduces a phase shift, such that the zero-order diffracted beam is cancelled, and if p<2λ, only the second spatial-frequency-multiple will be present. This should enable quasiperiodic patterns such as variable-period gratings and Fresnel zone plates to be ’’doubled’’.