Abstract: According to Thompson’s principle of similarity, the area of an object should be proportional to its length squared. However, leaf area–length data of some plants have been demonstrated not to follow the principle of similarity. We explore the reasons why the leaf area–length allometry deviates from the principle of similarity and examine whether there is a general model describing the relationship among leaf area, width and length. We sampled more than 11,800 leaves from six classes of woody and herbaceous plants and tested the leaf area–length allometry. We compared six mathematical models based on root-mean-square error as the measure of goodness-of-fit. The best supported model described a proportional relationship between leaf area and the product of leaf width and length (i.e., the Montgomery model). We found that the extent to which the leaf area–length allometry deviates from the principle of similarity depends upon the extent of variation of the ratio of leaf width to length. Estimates of the parameter of the Montgomery model ranged between 1/2, which corresponds to a triangular leaf with leaf length as its height and leaf width as its base, and π/4, which corresponds to an elliptical leaf with leaf length as its major axis and leaf width as its minor axis, for the six classes of plants. The narrow range in practice of the Montgomery parameter implies an evolutionary stability for the leaf area of large-leaved plants despite the fact that leaf shapes of these plants are rather different.

Abstract: Exploring self-assembled nanostructures with controllable architectures has been a central theme in nanoscience and nanotechnology because of the tantalizing perspective of directly integrating such bottom-up nanostructures into functional devices. Here, the growth of kinked single-crystal In2O3 nanostructures consisting of a nanocone base and a nanowire tip with an epitaxial and defect-free transition is demonstrated for the first time. By tailoring the growth conditions, a reliable switching of the growth direction from [111] to [110] or [112] is observed when the Au catalyst nanoparticles at the apexes of the nanocones shrink below approximate to 100 nm. The natural formation of kinked nanoarchitectures at constant growth pressures is related to the size-dependent free energy that changes for different orientations of the nanowires. The results suggest that the mechanism of forming such kinked nanocone-nanowire nanostructures in well-controlled growth environment may be universal for a wide range of functional materials.