Deployable motion of rotational sliceforms

A rotational sliceform (RS) forms a stiff, ringlike array of intersecting planar slices. Removing a few slices and disconnecting the ends of an RS enables the incomplete array to be collapsed scissorlike into a compact stack; it can be expanded smoothly as far as the original incomplete configuration, but not beyond. Its structured architecture, coupled to apparent mechanistic motion and a natural self-locking ability, expresses equivalently a novel deployable metamaterial, and we set out to determine its natural limits of motion for symmetrical and asymmetrical RS architectures. We first reconceptualize the RS as an array of plane-faced pyramidal cells bounded by rigid slices of zero thickness. The minimum articulation range from all cells is shown to set an upper bound on the range of motion of an incomplete RS, specifically, that symmetrical architectures can collapse fully while asymmetrical cannot and that expansion always stops at the design configuration. We also find that planar rotation of slices is not possible without distorting the original intersections. Each slice is then permitted to kink out-of-plane while preserving the initial geometry of each cell, in order to marshal compatible rotations of now compliant slices. Our analysis then reliably captures the deployment features: the minimum collapsed state, the degree of slice deformation as they rotate, and the limit of expansion.