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We demonstrate the emergence of complexity from remarkably simple and ubiquitous systems:  draining thin-film suspensions exhibiting a striking transition between two classes of self-organizing patterns. Vertical channels form when attractive forces lead to transient gelation, while horizontal bands result from granular mixtures. We propose an explanation whereby the generic physical mechanisms require only the existence of viscous and excluded-volume couplings among the particles, solvent, and substrate. System-specific, small inhomogeneities trigger large-scale pattern formation, through collective dynamics, where jamming plays a crucial role. Our results shed light on emergent complexity in bio- and geophysical processes and have implications for coatings and food industries. [ABSTRACT FROM AUTHOR]