The geometry of cutting and shuffling: An outline of possibilities for piecewise isometries

Cutting and shuffling is emerging as an alternative mixing mechanism for fluids and granular matter beyond the well established stretching and folding. Dynamical systems and chaos theory provided a foundation for stretching and folding which has led to applications ranging from microfluidic devices and physiological scales to many engineering and Earth science scales. Likewise, the literature of piecewise isometries (PWIs) provides a similar grounding for cutting and shuffling mechanisms. We start with one-dimensional interval exchange transformations (IETs), which are the only way to cut and shuffle in one dimension, and review and extend previous studies, connecting them in a coherent way. We introduce the concept of time-continuous piecewise isometries, i.e. PWIs that can be performed on solid bodies in a time continuous manner, without solids overlapping or the domain needing to be deformed or extended. PWIs with this property are easier to implement in experiment and applications, as we demonstrate through their connection to mixing in spherical granular tumblers and “twisty puzzles,” such as the spherical version of the Rubik's cube.