Cutting and shuffling a hemisphere: non-orthogonal axes

TL;DR

This paper explores how non-orthogonal axes in a hemispherical shell's cutting-and-shuffling process affect mixing, revealing new periodic protocols and enhanced mixing due to broken symmetries.

Contribution

It introduces a new computational approach and extends PWI theory to non-orthogonal axes, uncovering novel dynamics and mixing behaviors.

Findings

Non-orthogonal axes improve overall mixing.

Arnold tongues influence mixing protocols.

Periodic protocols form polygonal tilings.

Abstract

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for non-orthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Non-orthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the non-orthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.