Hexagon formation as a compromise between squares and stripes in vertically vibrated granular layers

We propose a spatially continuous and temporally discrete model for pattern formation in vertically vibrated granular layers. The grain transfer and t he grain mobility transitions are introduced qualitatively, but explicitly. This model reproduces experimentally observed f/2 patterns, including squa res in high-mobility conditions arid stripes in low-mobility conditions. He xagons are obtained when a high-mobility condition and a low-mobility condi tion are alternately applied. We discuss this hexagon formation in connecti on with the formation of squares and stripes. In addition to the experiment ally observed localized oscillon structures. this model also exhibits local ized hexagonal structures, These localized hexagonal structures grow by nuc leating oscillons of equal phase.