A mathematical model is proposed to describe the formation of orientation columns in mammalian visual cortex. The model is similar in concept to that proposed for ocular dominance column formation (Swindale 1980), the essential difference being that orientation is a vector rather than a scalar variable. It is assumed that initially orientation selectivity is weak and randomly distributed, and that selectivity develops in such a way that the orientation preferences of neurons less than about 200 microns apart tend to change in a similar direction, whereas the preferences of cells further apart tend to develop in opposite directions. No hypotheses are made about the anatomical or physiological basis of these interactions, and it is not necessary to assume that they are the result of environmental stimulation, as with existing models for the development of orientation selectivity (see, for example, von der Malsburg, 1973). The model reproduces the experimental data on orientation columns: roughly linear sequences of orientation change are produced, and these alternate unpredictably between clockwise and anticlockwise directions of change. Continuous sequences may span several 180 degrees cycles of rotation. The sequences are generally smooth, but abrupt discontinuities of up to 90 degrees also occur. The iso-orientation domains for large orientation ranges (60-90 degrees) are periodically spaced branching stripes that resemble those demonstrated in animals by the 2-deoxyglucose technique. The domains for narrower orientation ranges are periodically spaced but are more irregular in shape, though sometimes thin and elongated. The model makes a number of predictions that can be tested experimentally. Of particular interest are the discontinuities in the orientation sequences: these should be distributed with a spacing roughly equal to, or half, that of the iso-orientation domains. Each should be surrounded by one or two complete sets of iso-orientation domains, and each may be associated with regions where cells are not orientation selective. These regions may be more extensive in younger animals, when the columns are at an intermediate stage of formation, and less numerous where the columns run parallel and unbranched over large areas.