Abstract
Helices are amongst the most common structures in nature and in some cases, such as tethered plant tendrils, a more complex but related shape, the hemihelix forms. In its simplest form it consists of two helices of opposite chirality joined by a perversion. A recent, simple experiment using elastomer strips reveals that hemihelices with multiple reversals of chirality can also occur, a richness not anticipated by existing analyses. Here, we show through analysis and experiments that the transition from a helical to a hemihelical shape, as well as the number of perversions, depends on the height to width ratio of the strip's cross-section. Our findings provides the basis for the deterministic manufacture of a variety of complex three-dimensional shapes from flat strips.
Highlights
Nature abounds with complex, three-dimensional shapes [1,2]
The chirality can switch during growth as noted by Asa Gray [7] and Darwin [8] in their studies of plant tendrils
They noted that as a growing plant tendril circumnutates it can attach to another object and being fixed at both ends, its chirality reverses in between to maintain its topology as it continues to grow [9,10]
Summary
Three-dimensional shapes [1,2]. Of these, the helix and spiral are amongst the most ubiquitous, often emerging during growth from initially straight or flat 2-D configurations. For the case of small aspect ratios h=w, the formation of hemihelices with multiple reversal of chirality is observed in the experiments during release These complex 3D shapes can be captured by studying perturbed states of the systems in a small neighborhood of the straight configuration [13,29,30] (see File S1 for details). If h=w further increases, the growth rate is maximum for nv, so that the formation of a helix is expected These analytical predictions are fully consistent with the experimental results shown, where it is clearly shown that the mode number monotonically decreases as a function of h=w. The red line defines the geometric transition between hemihelices and helices and the shaded region indicates where hemihelices form
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