A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo

Abstract

Many cross-sectional shapes of plants have been found to approximate a superellipse rather than an ellipse. Square bamboos, belonging to the genus Chimonobambusa (Poaceae), are a group of plants with round-edged square-like culm cross sections. The initial application of superellipses to model these culm cross sections has focused on Chimonobambusa quadrangularis (Franceschi) Makino. However, there is a need for large scale empirical data to confirm this hypothesis. In this study, approximately 750 cross sections from 30 culms of C. utilis were scanned to obtain cross-sectional boundary coordinates. A superellipse exhibits a centrosymmetry, but in nature the cross sections of culms usually deviate from a standard circle, ellipse, or superellipse because of the influences of the environment and terrain, resulting in different bending and torsion forces during growth. Thus, more natural cross-sectional shapes appear to have the form of a deformed superellipse. The superellipse equation with a deformation parameter (SEDP) was used to fit boundary data. We find that the cross-sectional shapes (including outer and inner rings) of C. utilis can be well described by SEDP. The adjusted root-mean-square error of SEDP is smaller than that of the superellipse equation without a deformation parameter. A major finding is that the cross-sectional shapes can be divided into two types of superellipse curves: hyperellipses and hypoellipses, even for cross sections from the same culm. There are two proportional relationships between ring area and the product of ring length and width for both the outer and inner rings. The proportionality coefficients are significantly different, as a consequence of the two different superellipse types (i.e., hyperellipses and hypoellipses). The difference in the proportionality coefficients between hyperellipses and hypoellipses for outer rings is greater than that for inner rings. This work informs our understanding and quantifying of the longitudinal deformation of plant stems for future studies to assess the influences of the environment on stem development. This work is also informative for understanding the deviation of natural shapes from a strict rotational symmetry.

Half-leaf width symmetric distribution reveals buffering strategy of Cunninghamia lanceolata

“…The vascular tapering patterns revealed the buffering strategy is helpful to understand the hydrodynamical, biomechanical and geometrical design of plants based on the application of resource distribution theory under global warming trends [ 3 ]. This is mostly confirmed in roots, trunks, and branches of broad-leaved species, such as bamboo [ 4 – 7 ]. However, leaves acting as a transfer with external water vapor exchange constitute a substantial (up to 60%) part of the resistance to water flow through plants, and thus influence transpiration, photosynthesis, and productivity [ 8 ].…”

“…Most studies have focused on which leaf vein tapering (e.g., midrib or veinlet) allows to decrease the path resistance and increase hydraulic capacity relative to construction costs [ 14 – 16 ]. But the leaf width shows tremendous variation based on similar vein tapering mechanisms [ 7 , 17 , 18 ]. Significantly, leaf width contributes more constraints to hydraulic design, because the maximum mesophyll hydraulic pathway is determined by leaf width [ 14 , 19 ].…”

Evidence That Supertriangles Exist in Nature from the Vertical Projections of Koelreuteria paniculata Fruit

“…We refer to Equation (2) as the Gielis equation (GE) for convenience hereinafter. The GE is derived from the superellipse equation, which can generate diamonds, ellipses, rectangles, and transitional shapes between these classical geometries [4,5]. The GE has been demonstrated to reflect the natural shapes of bamboo leaves [6,7], cross sections of plant stems [5,8], avian eggs [9], and ginkgo seeds [3].…”

“…The GE is derived from the superellipse equation, which can generate diamonds, ellipses, rectangles, and transitional shapes between these classical geometries [4,5]. The GE has been demonstrated to reflect the natural shapes of bamboo leaves [6,7], cross sections of plant stems [5,8], avian eggs [9], and ginkgo seeds [3]. However, for radially symmetrical objects with a finite number of multiple axes of symmetry, such as some species of sea stars, the GE cannot validly describe their shapes.…”

Magnetization Reversal in Concave Iron Nano-Superellipses

“…Here we investigate the influence of concave deformations of the edges of a square nanodot, resulting in so-called superellipses, sometimes also called subellipses. The formula of such superellipses is |x/a| n + |y/b| n = 1, resulting in a diamond for n = 1 [28]. Here, a and b were chosen to be identical, enabling producing a square for n = 1, a circle for n = 2, superellipses with convex edges for n > 1, and superellipses with concave shapes (also called subellipses) for n < 1.…”