Figure 1

An illustration of Euclidean and non-Euclidean strips. Three types of unit cells are depicted, and above each such unit cell we display strips obtained by stacking multiple copies of them one on top of the other. (a) A rectangular block: the side length along the $v$ direction is constant. (b) When multiple blocks are stacked one on top of the other along the $v$ direction a straight thin and narrow strip is obtained. (c) A block whose length along the $v$ direction varies linearly with the $u$ coordinate. (d) When multiple blocks are stacked the resulting strip remains planar, yet curves in the plane to accommodate the length variation. (e) A block displaying a nonlinear length variation. When such blocks are stacked along the $v$ direction the resulting strip must curve out of plane in order to accommodate the nonlinear length variation. There are many different configurations compatible with the prescribed lengths. Two such configurations are depicted: (f) a catenoid-like configuration and (g) a helicoidal configuration. In the limit of vanishing unit cell length, the length variation within the unit cell maps onto a metric for the strip. A nonlinear length variation along the $v$ direction yields a non-Euclidean metric.Reuse & Permissions

Figure 2

An illustration of a nanometric non-Euclidean strip. Silicon doped carbon nanoribbon. (a) A single carbon base unit cell in which the right-most and left-most atoms were replaced by silicon atoms. The silicon-carbon bond length ($L_1$) is roughly 15% longer than the carbon-carbon bond length ($L_0$). When a single unit cell is considered, its equilibrium configuration lays in the plane. (b) When six such unit blocks are connected, the resulting ribbon will bend out of plane to accommodate the length variation across its width, similarly to the strip in Fig. 1f. The length scale of the Gaussian curvature associated with the silicon doping-induced elongation is of the order of the width of the strip. Therefore, while the strip depicted in (b) is a non-Euclidean plate, it cannot be considered narrow. The figures above are only illustrations; for further details see [11].Reuse & Permissions

Figure 3

Skeletonization of a strip. An illustration of a single period of a minimal surface. Along the midcurve of the strip $K=-1$ and $k_g=0.75$. (a) The full surface obtained by the explicit solutions (27) and (28). (b) The “skeleton” of the surface composed from the midcurve (blue) solid curve and tangent vectors $\partial {}_1\mathit{f}$ (black) arrows using the explicit formulas (B1) and (24). (c) When studying long and narrow surfaces it is convenient to display only the midcurve and the end points of each period.Reuse & Permissions

Figure 4

Four periods of the midcurve of four almost minimal surfaces. Along the midcurve of the strips $K=-1$ and the geodesic curvature is (a) $k_g=0.1$, (b) $k_g=0.4$, (c) $k_g=0.7$, and (d) $k_g=1$. Each period is plotted in a different color and is separated from the next period by a filled (black) circle. The four curves are not to scale.Reuse & Permissions

Figure 5

Twenty periods of the midcurve for the same parameter as in Fig. 4. Plotting many periods exemplifies not only that the midcurves are bounded in space, but also that they lie on a surface of revolution. The midcurve is plotted as a solid (blue) line, (black) filled circles locate the periods end points, and the bolded (red) line corresponds to a single period. Note also that all period end points lie on a circle. For graphical reasons the curves are not to scale.Reuse & Permissions

Figure 6

Minimal surfaces. The surfaces were obtained by explicit integration of the Gauss-Weingarten equations (30) and (31). The parameters of the strips are the same as for the previous figures; along the midcurve of the strips $K=-1$ and the geodesic curvature is (a) $k_g=0.1$, (b) $k_g=0.4$, (c) $k_g=0.7$, and (d) $k_g=1$. Strips colors correspond to longitudinal strain relative to the pseudospherical metric (14), where dark (blue) corresponds to no strain and the lighter (red) colors near the edge correspond to the maximal strain magnitude of $5\times {10}^{-4}$.Reuse & Permissions

Figure 7

Sensitivity to metric variations. A variation of $1/20$ in the geodesic curvature of the midcurve $k_g$ yields very strong morphological changes. All three strips are minimal surfaces calculated as described in (26) and have Gaussian curvature $K=-1$ along their midcurve. One period is plotted for each strip. Colors (same as in Fig. 6) are given for visual purposes.Reuse & Permissions

Figure 8

Translationally invariant configurations of various pitch. $H^2$ as function of $x_1$ for four different embeddings of the pseudospherical metric (14) ($K=-1$ everywhere) for the parameters ${\kappa }_g=0.4$, $w=0.4$, and $L=4\pi$. In all cases $H=0$ along the midcurve and the configurations differ by the value of $\tau$. The inset shows the solutions to $\tau =0,-7/25,-3/5,-15/17$ (left-to-right) next to their corresponding curves.Reuse & Permissions

Figure 9

The $\beta =0$ limit for finite strips. As $\beta \to 0$ we expect the solutions obtained to converge to the $\beta =0$ solution. Optimizing for the location within the midcurve of a $\beta \ne 0$ solution (the phase $\theta$) yields a strip segment very similar to that obtained for $\beta =0$. (a) One period (of length $\lambda =2\pi /\beta$) of the midcurve for a strip with metric (14) and ${\kappa }_g=0.1$ along with a strip segment of length $\lambda /10$. Optimization with respect to the phase finds which segment along the period has the smallest Willmore energy. (b) The strip segment enlarged. (c) The $\tau =0$ continuously symmetric strip for the same values of length and width.Reuse & Permissions

Figure 10

Typical topology of the two families of asymptotic curves for a minimal surface of the type described in Sec. 5. At every point two characteristic curves from distinct families intersect. (a) The straight bold (black) line is the midcurve along which initial data values are prescribed. One characteristic curve that is tangent to the midcurve is also bolded (blue). All characteristic curves from the same family which lay to the left of this curve do not intersect the midcurve, and therefore are uninfluenced by the initial data. Characteristic curves of the same family which lay to the right of this curve intersect the midcurve twice and may generate a conflict of initial data. (b) A blowup of a section of the strip. The shaded region marks the domain of influence of initial data given along the midcurve. Prescribing additional partial data along the bolded dashed (blue) line is needed in order to determine the solution uniquely in the unshaded region.Reuse & Permissions