Cutting and shuffling a hemisphere: Nonorthogonal axes

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 nonorthogonal 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. Nonorthogonal 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 nonorthogonal-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.

Figure 1

Demonstration of the rotations that define the PWI, here $\alpha ={57}^{\circ }$, $\beta ={57}^{\circ }$, and $\gamma ={120}^{\circ }$. (a) The initial condition of the hemispherical shell (HS). (b) Rotation about the first axis ($z$ axis) by $\alpha$. (c) Separation of the portion above the equator and beginning of the rotation to reconstruct the HS. (d) Re-formed HS. (e)–(h) Repeat of (a)–(d) about the second horizontal axis.

Figure 2

The PWI mapping $M_{{57}^{\circ },{57}^{\circ },{120}^{\circ }}$ (shown from the negative $y$ axis) (a) cuts the domain into four atoms, $P_1,P_2,P_3,P_4$, and (b) rearranges them according to the rotation procedure in Sec. 2. Cutting lines ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are the red (unbroken) and black (broken) arcs in (a). Cutting lines in (b) are formed by the images of $\partial S$ (broken blue arc) and ${\mathcal{D}}_1$ (unbroken red arc). Angles $\alpha ,\beta ,\gamma$ specify the size and orientation of the atoms. Angle ${\gamma }^{\prime }$ is the spherical angle between cutting planes. (c) The union of cutting lines (red and blue) when $M_{\alpha ,\beta ,\gamma }$ is applied 20 000 times.

Figure 3

Iterating the $({90}^{\circ },{90}^{\circ },{60}^{\circ })$ map. (a) The cutting line structure approaches the structure of the exceptional set as the number of iterations increases. Circular regions appear around periodic points. We use $\varepsilon =0.01$ for $1\le N\le 20$ and $\varepsilon =0.001$ for $200\le N\le 2\times {10}^4$. Larger values of $\varepsilon$ are used for small numbers of iterations to give a visible thickness to the few cutting lines that are present. (b) With a colored initial condition ($N=0$), the mixed part of the domain for $N=2\times {10}^4$ (gray region) closely matches regions in (a) that are densely cut, while circular unmixed regions correspond to open cells in (a).

Figure 4

The exceptional set ${\tilde{E}}_{\varepsilon ,N}$ for increasing $\gamma$ viewed from the negative $y$ axis with the $z$ axis pointing toward the top of the page $[\varepsilon =1\times {10}^{-4}$ and $N=2\times {10}^4$ except for $({90}^{\circ },{90}^{\circ },{90}^{\circ })$ which uses $\varepsilon =0.01$ to yield visible line thickness]. (a)–(d) $({90}^{\circ },{90}^{\circ },\gamma )$ and (e)–(h) $({57}^{\circ },{57}^{\circ },\gamma )$; $\gamma$ increases downward. One of the four period-2 cells for the $({90}^{\circ },{90}^{\circ },\gamma )$ protocols is labeled $A$. A period-4 cell is labeled $B$ in (d) and a period-3 cell is labeled $C$ in (e)–(h).

Figure 5

A resonance, or locally minimally mixing protocol, occurs for the $({57}^{\circ },{57}^{\circ },99.3^{\circ })$ protocol. $E$ is complete after only 160 iterations. $E$ forms a polygonal tiling, with large regular pentagons, small regular 32-gons, and irregular polygons (mostly quadrilaterals). Compare this simple cutting line structure to that of the densely packed $({57}^{\circ },{57}^{\circ },{100}^{\circ })$ protocol in Fig. 4 where $\gamma$ differs by only $0.7^{\circ }$.

Figure 6

Barriers to mixing occur in the $({90}^{\circ },\beta ,{90}^{\circ })$ case shown after $2\times {10}^4$ iterations of the initial condition for $\beta ={30}^{\circ },{60}^{\circ },{90}^{\circ }$. Although $\tilde{E}$, shown on the right, appears to indicate mixing throughout the domain (except, of course, the cells) based on the fractional coverage of $\tilde{E}$, the symmetry of the system resulting from orthogonal rotation axes creates a barrier to mixing between the left and right halves of the hemisphere.

Figure 7

Various initial conditions (left column) under $({60}^{\circ },{60}^{\circ },{90}^{\circ })$ and $({90}^{\circ },{60}^{\circ },{90}^{\circ })$ after $2\times {10}^4$ iterations show sensitivity of mixing to initial conditions. Initial condition varied in (a) vertical $z$ direction, (b) both $x$ (blue) and $z$ (yellow) directions, and (c) horizontal $x$ direction.

Figure 8

Fractional coverage, ${\mathrm{\Phi }}_{0.01,500}$, vs rotation angles for $\gamma ={90}^{\circ }$ to ${165}^{\circ }$ in ${15}^{\circ }$ increments after 500 iterations. Increments in $\alpha$ and $\beta$ are $0.{33}^{\circ }$. (a) For $\gamma ={90}^{\circ }$, reflection symmetry lines are shown as red ($\alpha +\beta ={180}^{\circ }$ and $\alpha =\beta$) and blue ($\alpha ={90}^{\circ }$ and $\beta ={90}^{\circ }$) dashed lines. Symmetry along the red dashed lines is maintained for nonorthogonal axes while the blue dashed symmetries are lost. (a)–(f) Black dashed curves mark the locations of Arnold tongues originating at $({60}^{\circ },0,\gamma )$ and $(0,{60}^{\circ },\gamma )$. Blue dashed curves mark Arnold tongues starting at $({90}^{\circ },{180}^{\circ },\gamma )$ and $({180}^{\circ },{90}^{\circ },\gamma )$. (c) The protocol $(65.5^{\circ },65.5^{\circ },{120}^{\circ })$ is labeled $A$ and lies on a dark band between Arnold tongue intersections. (d) The protocol labeled $B$ lies close to the intersection of the Arnold tongues from $({60}^{\circ },0,\gamma )$, $(0,{60}^{\circ },\gamma )$ (black dashed curves) and $({135}^{\circ },{180}^{\circ },\gamma )$, $({180}^{\circ },{135}^{\circ },\gamma )$ (green dashed curves). The short-dashed lines show the corresponding mirror Arnold tongues for long-dashed lines of the same color. The intersections of short-dashed and long-dashed lines are often locally resonant.

Figure 9

(a) Fractional coverage, ${\mathrm{\Phi }}_{0.01,500}$, for protocols along the $\alpha =\beta$ symmetry plane and varying axes angle, $\gamma$. The upper half, $\gamma \ge {90}^{\circ }$, corresponds with behavior along the $\alpha =\beta$ line in Fig. 8 and the lower half, $\gamma \le {90}^{\circ }$, corresponds with the $\alpha +\beta ={180}^{\circ }$ line in Fig. 8 due to the symmetries in Eqs. (A5) and (A6). (b) The same symmetry plane with resonant curves defined by Eqs. (9) and (10) for $j=1,2,3,7$. The boxed region is shown in Fig. 10. The protocol labeled $A$, $(65.5^{\circ },65.5^{\circ },{120}^{\circ })$, corresponds with Fig. 8. The protocol labeled $B$ near the intersection of the branches from $(\pi /5,\pi /5,0)$ and $(6\pi /7,6\pi /7,0)$ corresponds with $B$ in Fig. 8.

Figure 10

A detailed view of the region where the $\frac{\pi }{5}$ and $\frac{2\pi }{3}$ branches intersect marked by the dashed black box in Fig. 9. Some of the $\tilde{E}$ structures along these branches after $N=2\times {10}^4$ iterations are shown on the right with their locations in the protocol space marked by $\times$ for the $\pi /5$ branch ($2j+1=5$) and $\circ$ for the $2\pi /3$ branch ($2j+1=3$). The protocol labeled $A$, $(65.5^{\circ },65.5^{\circ },{120}^{\circ })$, corresponds with Figs. 8 and 9. $\varepsilon =0.01$ for $\gamma \le {108}^{\circ }$ and $\varepsilon =5\times {10}^{-4}$ for $\gamma \ge {109}^{\circ }$ for the structures on the right.

Figure 11

${\mathrm{\Phi }}_{0.01,2\times {10}^4}$ averaged across $0\le \alpha ,\beta \le {180}^{\circ }$ vs $\gamma$. $\mathrm{\Phi }$ increases on average for axis arrangements farther from orthogonal.

Figure 12

Curves of equal atom size [Eqs. (2, 3, 4, 5)] for $M_{\alpha ,\beta ,\gamma }$ where $\gamma ={120}^{\circ }$. The smallest atom, $P_i$, is labeled inside of each region between curves.