Remarkably persistent mixing and non-mixing regions (islands) are observed to coexist in a three-dimensional dynamical system where randomness is expected. The track of an x-ray opaque particle in a spherical shell half-filled with dry non-cohesive particles and periodically rotated about two axes reveals interspersed structures that are spatially complex and vary non-trivially with the rotation angles. The geometric skeleton of the structures forms from the subtle interplay between fluid-like mixing by stretching-and-folding, and solids mixing by cutting-and-shuffling, which is described by the mathematics of piecewise isometries. In the physical system, larger islands predicted by the cutting-and-shuffling model alone can persist despite the presence of stretching-and-folding flows and particle-collision-driven diffusion, while predicted smaller islands are not observed. By uncovering the synergy of simultaneous fluid and solid mixing, we point the way to a more fundamental understanding of advection driven mixing in materials with both solid and flowing regions.
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