Mixing due to cutting-and-shuffling is studied at a fundamental level using 2D mappings known as Piecewise Isometries (PWI) which can create beautiful mixing patterns. The PWI studied here splits a hemispherical shell (HS) into four curved triangular pieces that are rearranged to make a shuffled HS. Applying the PWI repeatedly (1, 20, and 20,000 times) to an initial condition (left) reveals circular regions devoid of mixing and regions that appear well-mixed (gray), the size of which determine how well the HS is mixed. PWI operations on initial conditions like those shown require highly parallel computation on a GPU to allow adequate repetition to resolve mixing patterns. The behavior of PWIs can be used to design efficient mixing systems. Funded by NSF CMMI 1435065.
Intricate mixing patterns emerge when a hemisphere (viewed from below) is repeatedly cut into four pieces and rearranged (shuffled). The hemisphere mixes in some regions (gray) but not in others. Funded by NSF CMMI 1435065.