Experiments reveal that an initially well mixed granular material composed of two distinct subclasses of particles, small and large or light and heavy, segregates radially into stable lobed patterns when rotated in various quasi two-dimensional, regular polygonal tumblers. The patterns are highly sensitive to the time periodic flow, which in turn depends critically on the fill fraction and container shape. Simulations of a simple model reproduce the observed segregation patterns. KAM regions in Poincaré plots of the velocity field used to model the flow attract smaller (denser) particles and their spatial symmetries mirror those of the segregation patterns, suggesting that competition between the driving forces for radial segregation (percolation and buoyancy) and those for chaotic mixing play a key role in determining the patterns.