Abstract 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.

中文翻译：

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切割和改组的几何：分段等距的可能性概述

摘要 切割和改组正在成为一种替代的流体和颗粒物质混合机制，超越了完善的拉伸和折叠。动力系统和混沌理论为拉伸和折叠提供了基础，这导致了从微流体设备和生理尺度到许多工程和地球科学尺度的应用。同样，分段等距 (PWI) 的文献为切割和改组机制提供了类似的基础。我们从一维区间交换变换 (IET) 开始，这是在一维中进行切割和洗牌的唯一方法，并回顾和扩展以前的研究，以连贯的方式将它们连接起来。我们引入了时间连续分段等距的概念，即可以在实体上以时间连续方式执行的 PWI，没有实体重叠或域需要变形或扩展。具有这种特性的 PWI 更容易在实验和应用中实现，正如我们通过它们与球形颗粒状不倒翁和“扭曲谜题”（例如魔方的球形版本）中的混合所展示的那样。