The question of how to prescribe desired global behaviours for a system of interconnected agents through the application of only simple and local interactions has both theoretical and practical significance. In recent years, numerous papers have appeared in the control systems literature on the reconfiguration and stabilisation of multivehicle formations. This study describes a unique kind of asymptotic behaviour; namely, stable periodic formations. It is shown how these can be generated when multiple non-honolonomic vehicles pursue one another under a cyclic interconnection topology known in the mathematics literature as `cyclic pursuit`. Herein, particular attention is provided to the case when the number of vehicles is even. Broadly put, our objective is to introduce a broadening of the study of multi-agent control that includes not only static but also periodic or, more generally, dynamic formations.