Mixing and the fractal geometry of piecewise isometries

Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, $E$, whose complement comprises nonmixing regions in the domain. Varying the parameters that define the PWI changes both the structure of $E$ as well as the degree of mixing the PWI produces, which begs the question of how to determine which parameters produce the best mixing. Motivated by mixing of yield stress materials, for example granular media, in physical systems, we use numerical simulations of PWIs on a hemispherical shell and examine how the fat fractal properties of $E$ relate to the degree of mixing for any particular PWI. We present numerical evidence that the fractional coverage of $E$ negatively correlates with the intensity of segregation, a standard measure for the degree of mixing, which suggests that fundamental properties of $E$ such as fractional coverage can be used to predict the effectiveness of a particular PWI as a mixing mechanism.

Figure 1

Bottom view of a hemispherical shell (HS) showing (a) the initial location of the pieces to be rearranged, $P_i$, and their corresponding boundaries comprising $\mathcal{D}$ (black and red arcs) and the domain boundary, $\partial S$ (blue arc) and (b) their subsequent locations after their rearrangement by one iteration of the PWI. Reproduced from Park et al. [25] with permission from AIP Publishing.

Figure 2

PWI operation on the HS for $({\theta }_z,{\theta }_x)=({45}^{\circ },{45}^{\circ })$. Dashed lines indicate the two rotation axes. (a) Initial condition. (b) Rotation and cut about the $z$ axis by ${\theta }_z$. (c) Rotation and cut about the $x$ axis by ${\theta }_x$. (d) New state after one iteration. Adapted from Park et al. [25] with permission from AIP Publishing.

Figure 3

Bottom view of (a) initial condition of seeded colors on the HS, (b) mixing on the HS for the $({54}^{\circ },{54}^{\circ }), ({75}^{\circ },{75}^{\circ })$, and $({30}^{\circ },{15}^{\circ })$ protocols after $m=150$ iterations, and (c) $E$ for corresponding protocols obtained by tracking the respective $\mathcal{D}$ for 5000 iterations.

Figure 4

Magnifying $E$ for the $({54}^{\circ },{54}^{\circ })$ protocol by $8\times$ to various scales. Each magnification, from (a) to (b), from (b) to (c), and from (c) to (d), of the red box reveals circular cells at progressively finer scales.

Figure 5

Visualizing $E$ with $m=5000$ for ${\theta }_z$ and ${\theta }_x\in [{15}^{\circ },{90}^{\circ }]$ varying in ${15}^{\circ }$ increments (bottom view). Reproduced from Park et al. [25] with permission from AIP Publishing.

Figure 6

Progression of the approximation of $E$ as a function of the number of protocol iterations $m$ (bottom view). Note that $E$ for the $({15}^{\circ },{15}^{\circ })$ protocol at $m=1000$ is not fully developed.

Figure 7

(a) An unfolded isocube half whose resolution for a quarter of one face is $2\times 2$ (top) and $2^3\times 2^3$ (bottom). The total number of boxes is $N=12\times 2^n\times 2^n$, since there are 12 quarter faces (each quarter face is outlined with bold lines). (b) Isocube half in (a) mapped to the HS as viewed from the bottom.

Figure 8

Measuring the fraction of coverage of $E$ for the $({90}^{\circ },{45}^{\circ })$ protocol at resolutions of $n\in \{3,4,5,6\}$. Boxes that contain portions of $E$ are black (bottom view).

Figure 9

Fitting the measured values of ${\mathrm{\Phi }}_n$ for various protocols. Corresponding $E$ shown on the right.

Figure 10

No obvious correlation between the asymptotic coverage ${\mathrm{\Phi }}_{\infty }$ and exterior dimension $d_{\text{ext}}$ exists for the 4095 protocols examined.

Figure 11

Intensity of segregation $\mathcal{I}$ versus iteration number for six sample protocols; corresponding $E$ are shown on the right.

Figure 12

Distribution of $\sigma \left(\mathcal{I}\right)$ for all 4095 protocols over a range of $s=50$ iterations at different $m$. There is some overlap between distributions for different ranges of iterations; the distribution of later iterations are plotted over earlier ones.

Figure 13

Comparing $\mathcal{I}$ to ${\mathrm{\Phi }}_{\infty }$ for all 4095 protocols at iterations $m=25, {10}^2$, and ${10}^3$. At high $m, \mathcal{I}$ and ${\mathrm{\Phi }}_{\infty }$ appear to be negatively, linearly correlated.

Figure 14

Comparing $E$ for $({\theta }_1,{\theta }_2)$ and $({\theta }_2,{\theta }_1)$ for two different pairs of ${\theta }_1$ and ${\theta }_2$ (bottom view). Middle column shows the image transpose of $E$ for $({\theta }_1,{\theta }_2)$, which is a reflection across $z=-x$ in the coordinate space. The image transpose of $E$ for $({\theta }_1,{\theta }_2)$ appears identical to $E$ for $({\theta }_2,{\theta }_1)$.

Figure 15

Comparing the progression towards $E$ for ${\theta }_1={75}^{\circ }$ and ${\theta }_2={45}^{\circ }$ (bottom view). Left column, progression for $({\theta }_1,{\theta }_2)$; middle column, image transpose of the left column; and right column, progression for $({\theta }_2,{\theta }_1)$. It is not apparent that the resulting $E$ from both will be the same.

Figure 16

Comparing $\mathcal{D}$ (top) and $\mathcal{D}\cup M^{-1}\left(\mathcal{D}\right)$ (bottom) for the $({75}^{\circ },{45}^{\circ })$ protocol (left column), their image transpose (middle column), and the progression towards $E$ for the $({45}^{\circ },{75}^{\circ })$ protocol at $m=1$ and 2 (bottom view).

Figure 17

$E$ for ${\theta }_z$ and ${\theta }_x\in [{45}^{\circ },{60}^{\circ }]$ varying in $3^{\circ }$ increments.

Figure 18

$E$ for ${\theta }_z$ and ${\theta }_x\in [{45}^{\circ },{50}^{\circ }]$ varying in $1^{\circ }$ increments.

Figure 19

$E$ for ${\theta }_z$ and ${\theta }_x\in [{45}^{\circ },{46}^{\circ }]$ varying in $0.2^{\circ }$ increments.