Computer simulations of a nonlinear planar system of first-order ODEs which we developed from a simple assumption that biological swarming is an outcome of aggregative behavior of the individuals in the swarm showed a surprising and novel outcome; the emergence of uniquely structured patterns that are intriguingly intricate, exquisite, symmetrical and regular. Some patterns look like flowers; others look like Spirograph curves and Guilloché patterns but with more intricate variations. Unlike Spirograph curves though, the swarm-induced patterns cannot be reproduced by any closed-form formula or by another pattern subjected to some resizing, translation, rotation and/or reflection.