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To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man‐made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non‐Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.
Plant leaves exhibit diverse shapes that enable them to utilize a light resource maximally. If there were a general parametric model that could be used to calculate leaf area for different leaf shapes, it would help to elucidate the adaptive evolutional link among plants with the same or similar leaf shapes. We propose a simplified version of the original Gielis equation (SGE), which was developed to describe a variety of object shapes ranging from a droplet to an arbitrary polygon. We used this equation to fit the leaf profiles of 53 species (among which, 48 bamboo plants, 5 woody plants, and 10 geographical populations of a woody plant), totaling 3310 leaves. A third parameter (namely, the floating ratio c in leaf length) was introduced to account for the case when the theoretical leaf length deviates from the observed leaf length. For most datasets, the estimates of c were greater than zero but less than 10%, indicating that the leaf length predicted by the SGE was usually smaller than the actual length. However, the predicted leaf areas approximated their actual values after considering the floating ratios in leaf length. For most datasets, the mean percent errors of leaf areas were lower than 6%, except for a pooled dataset with 42 bamboo species. For the elliptical, lanceolate, linear, obovate, and ovate shapes, although the SGE did not fit the leaf edge perfectly, after adjusting the parameter c, there were small deviations of the predicted leaf areas from the actual values. This illustrates that leaves with different shapes might have similar functional features for photosynthesis, since the leaf areas can be described by the same equation. The anisotropy expressed as a difference in leaf shape for some plants might be an adaptive response to enable them to adapt to different habitats.
Abstract Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called ‘biogeom’ that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree‐rings, and three growth‐rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. ‘biogeom’ includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth‐rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. ‘biogeom’ can quantify the intra‐ or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.
The leaf area, as an important leaf functional trait, is thought to be related to leaf length and width. Our recent study showed that the Montgomery equation, which assumes that leaf area is proportional to the product of leaf length and width, applied to different leaf shapes, and the coefficient of proportionality (namely the Montgomery parameter) range from 1/2 to π/4. However, no relevant geometrical evidence has previously been provided to support the above findings. Here, four types of representative leaf shapes (the elliptical, sectorial, linear, and triangular shapes) were studied. We derived the range of the estimate of the Montgomery parameter for every type. For the elliptical and triangular leaf shapes, the estimates are π/4 and 1/2, respectively; for the linear leaf shape, especially for the plants of Poaceae that can be described by the simplified Gielis equation, the estimate ranges from 0.6795 to π/4; for the sectorial leaf shape, the estimate ranges from 1/2 to π/4. The estimates based on the observations of actual leaves support the above theoretical results. The results obtained here show that the coefficient of proportionality of leaf area versus the product of leaf length and width only varies in a small range, maintaining the allometric relationship for leaf area and thereby suggesting that the proportional relationship between leaf area and the product of leaf length and width broadly remains stable during leaf evolution.
This book focuses on the origin of the Gielis curves, surfaces and transformations in the plant sciences. It is shown how these transformations, as a generalization of the Pythagorean Theorem, play an essential role in plant morphology and development. New insights show how plants can be understood as developing mathematical equations, which opens the possibility of directly solving analytically any boundary value problems (stress, diffusion, vibration...) . The book illustrates how form, development and evolution of plants unveil as a musical symphony. The reader will gain insight in how the methods are applicable in many divers scientific and technological fields.
Abstract The relationship between spatial density and size of plants is an important topic in plant ecology. The self‐thinning rule suggests a −3/2 power between average biomass and density or a −1/2 power between stand yield and density. However, the self‐thinning rule based on total leaf area per plant and density of plants has been neglected presumably because of the lack of a method that can accurately estimate the total leaf area per plant. We aimed to find the relationship between spatial density of plants and total leaf area per plant. We also attempted to provide a novel model for accurately describing the leaf shape of bamboos. We proposed a simplified Gielis equation with only two parameters to describe the leaf shape of bamboos one model parameter represented the overall ratio of leaf width to leaf length. Using this method, we compared some leaf parameters (leaf shape, number of leaves per plant, ratio of total leaf weight to aboveground weight per plant, and total leaf area per plant) of four bamboo species of genus Indocalamus Nakai ( I. pedalis (Keng) P.C. Keng, I. pumilus Q.H. Dai and C.F. Keng, I. barbatus McClure, and I. victorialis P.C. Keng). We also explored the possible correlation between spatial density and total leaf area per plant using log‐linear regression. We found that the simplified Gielis equation fit the leaf shape of four bamboo species very well. Although all these four species belonged to the same genus, there were still significant differences in leaf shape. Significant differences also existed in leaf area per plant, ratio of leaf weight to aboveground weight per plant, and leaf length. In addition, we found that the total leaf area per plant decreased with increased spatial density. Therefore, we directly demonstrated the self‐thinning rule to improve light interception.
A novel class of supershaped dielectric lens antennas, whose geometry is described by the three‐dimensional (3D) Gielis’ formula, is introduced and analysed. To this end, a hybrid modelling approach based on geometrical and physical optics is adopted in order to efficiently analyse the multiple wave reflections occurring within the lens and to evaluate the relevant impact on the radiation properties of the antenna under analysis. The developed modelling procedure has been validated by comparison with numerical results already reported in the literature and, afterwards, applied to the electromagnetic characterisation of Gielis’ dielectric lens antennas with shaped radiation pattern. Furthermore, a dedicated optimisation algorithm based on quantum particle swarm optimisation has been developed for the synthesis of 3D supershaped lens antennas with single feed, as well as with beamforming capabilities.
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.
Abstract The size and shape of plant leaves change with growth, and an accurate description of leaf shape is crucial for describing plant morphogenesis and development. Bilateral symmetry, which has been widely observed but poorly examined, occurs in both dicot and monocot leaves, including all nominated bamboo species (approximately 1,300 species), of which at least 500 are found in China. Although there are apparent differences in leaf size among bamboo species due to genetic and environmental profiles, bamboo leaves have bilateral symmetry with parallel venation and appear similar across species. Here, we investigate whether the shape of bamboo leaves can be accurately described by a simplified Gielis equation, which consists of only two parameters (leaf length and shape) and produces a perfect bilateral shape. To test the applicability of this equation and the occurrence of bilateral symmetry, we first measured the leaf length of 42 bamboo species, examining >500 leaves per species. We then scanned 30 leaves per species that had approximately the same length as the median leaf length for that species. The leaf‐shape data from scanned profiles were fitted to the simplified Gielis equation. Results confirmed that the equation fits the leaf‐shape data extremely well, with the coefficients of determination being 0.995 on average. We further demonstrated the bilateral symmetry of bamboo leaves, with a clearly defined leaf‐shape parameter of all 42 bamboo species investigated ranging from 0.02 to 0.1. This results in a simple and reliable tool for precise determination of bamboo species, with applications in forestry, ecology, and taxonomy.
During the past decades, the poration of cell membrane induced by pulsed electric fields has been widely investigated. Since the basic mechanisms of this process have not yet been fully clarified, many research activities are focused on the development of suitable theoretical and numerical models. To this end, a nonlinear, nonlocal, dispersive, and space-time numerical algorithm has been developed and adopted to evaluate the transmembrane voltage and pore density along the perimeter of realistic irregularly shaped cells. The presented model is based on the Maxwell's equations and the asymptotic Smoluchowski's equation describing the pore dynamics. The dielectric dispersion of the media forming the cell has been modeled by using a general multirelaxation Debye-based formulation. The irregular shape of the cell is described by using the Gielis' superformula. Different test cases pertaining to red blood cells, muscular cells, cell in mitosis phase, and cancer-like cell have been investigated. For each type of cell, the influence of the relevant shape, the dielectric properties, and the external electric pulse characteristics on the electroporation process has been analyzed. The numerical results demonstrate that the proposed model is an efficient numerical tool to study the electroporation problem in arbitrary-shaped cells.
There is convincing evidence for a scaling relationship between leaf dry weight (DW) and leaf surface area (A) for broad-leaved plants, and most estimates of the scaling exponent of DW vs. A are greater than unity. However, the scaling relationship of leaf fresh weight (FW) vs. A has been largely neglected. In the present study, we examined whether there is a statistically strong scaling relationship between FW and A and compared the goodness of fit to that of DW vs. A. Between 250 and 520 leaves from each of 12 bamboo species within 2 genera (Phyllostachys and Pleioblastus) were investigated. The reduced major axis regression protocols were used to determine scaling relationships. The fit for the linearized scaling relationship of FW vs. A was compared with that of DW vs. A using the coefficient of determination (i.e., r2). A stronger scaling relationship between FW and A than that between DW and A was observed for each of the 12 bamboo species investigated. Among the 12 species examined, five had significantly smaller scaling exponents of FW vs. A compared to those of DW vs. A; only one species had a scaling exponent of FW vs. A greater than that of DW vs. A. No significant difference between the two scaling exponents was observed for the remaining 6 species. Researchers conducting future studies might be well advised to consider the influence of leaf fresh weight when exploring the scaling relationships of foliar biomass allocation patterns.
The principle of similarity (Thompson, 1917) states that the weight of an organism follows the 3/2-power law of its surface area and is proportional to its volume on the condition that the density is constant. However, the allometric relationship between leaf weight and leaf area has been reported to greatly deviate from the 3/2-power law, with the irregularity of leaf density largely ignored for explaining this deviation. Here, we choose 11 bamboo species to explore the allometric relationships among leaf area (<i>A</i>), density (ρ), length (<i>L</i>), thickness (<i>T</i>), and weight (<i>W</i>). Because the edge of a bamboo leaf follows a simplified two-parameter Gielis equation, we could show that <i>A</i> ∝ <i>L</i><sup>2</sup> and that <i>A</i> ∝ <i>T</i><sup>2</sup>. This then allowed us to derive the density-thickness allometry ρ ∝ <i>T</i><sup><i>b</i></sup> and the weight-area allometry <i>W</i> ∝ <i>A</i><sup>(b+3)/2</sup> ≈ <i>A</i><sup>9/8</sup>, where <i>b</i> approximates -3/4. Leaf density is strikingly negatively associated with leaf thickness, and it is this inverse relationship that results in the weight-area allometry to deviate from the 3/2-power law. In conclusion, although plants are prone to invest less dry mass and thus produce thinner leaves when the leaf area is sufficient for photosynthesis, such leaf thinning needs to be accompanied with elevated density to ensure structural stability. The findings provide the insights on the evolutionary clue about the biomass investment and output of photosynthetic organs of plants. Because of the importance of leaves, plants could have enhanced the ratio of dry material per unit area of leaf in order to increase the efficiency of photosynthesis, relative the other parts of plants. Although the conclusion is drawn only based on 11 bamboo species, it should also be applicable to the other plants, especially considering previous works on the exponent of the weight-area relationship being less than 3/2 in plants.
Leaf shape is closely related to economics of leaf support and leaf functions, including light interception, water use, and CO2 uptake, so correct quantification of leaf shape is helpful for studies of leaf structure/function relationships. There are some extant indices for quantifying leaf shape, including the leaf width/length ratio (W/L), leaf shape fractal dimension (FD), leaf dissection index, leaf roundness index, standardized bilateral symmetrical index, etc. W/L ratio is the simplest to calculate, and recent studies have shown the importance of the W/L ratio in explaining the scaling exponent of leaf dry mass vs. leaf surface area and that of leaf surface area vs. leaf length. Nevertheless, whether the W/L ratio could reflect sufficient geometrical information of leaf shape has been not tested. The FD might be the most accurate measure for the complexity of leaf shape because it can characterize the extent of the self-similarity and other planar geometrical features of leaf shape. However, it is unknown how strongly different indices of leaf shape complexity correlate with each other, especially whether W/L ratio and FD are highly correlated. In this study, the leaves of nine Magnoliaceae species (>140 leaves for each species) were chosen for the study. We calculated the FD value for each leaf using the box-counting approach, and measured leaf fresh mass, surface area, perimeter, length, and width. We found that FD is significantly correlated to the W/L ratio and leaf length. However, the correlation between FD and the W/L ratio was far stronger than that between FD and leaf length for each of the nine species. There were no strong correlations between FD and other leaf characteristics, including leaf area, ratio of leaf perimeter to area, fresh mass, ratio of leaf fresh mass to area, and leaf roundness index. Given the strong correlation between FD and W/L, we suggest that the simpler index, W/L ratio, can provide sufficient information of leaf shape for similarly-shaped leaves. Future studies are needed to characterize the relationships among FD and W/L in leaves with strongly varying shape, e.g., in highly dissected leaves.
The electromagnetic analysis of a special class of 3D dielectric lens antennas is described in detail. This new class of lens antennas has a geometrical shape defined by the three-dimensional extension of Gielis’ formula. The analytical description of the lens shape allows the development of a dedicated semianalytical hybrid modeling approach based on geometrical tube tracing and physical optic. In order to increase the accuracy of the model, the multiple reflections occurring within the lens are also taken into account.
Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.
Gielis curves and surfaces can describe a wide range of natural shapes and they have been used in various studies in biology and physics as descriptive tool. This has stimulated the generalization of widely used computational methods. Here we show that proper normalization of the Levenberg-Marquardt algorithm allows for efficient and robust reconstruction of Gielis curves, including self-intersecting and asymmetric curves, without increasing the overall complexity of the algorithm. Then, we show how complex curves of k-type can be constructed and how solutions to the Dirichlet problem for the Laplace equation on these complex domains can be derived using a semi-Fourier method. In all three methods, descriptive and computational power and efficiency is obtained in a surprisingly simple way.
Abstract The shape of leaf laminae exhibits considerable diversity and complexity that reflects adaptations to environmental factors such as ambient light and precipitation as well as phyletic legacy. Many leaves appear to be elliptical which may represent a ‘default’ developmental condition. However, whether their geometry truly conforms to the ellipse equation (EE), i.e., ( x/a ) 2 + ( y/b ) 2 = 1, remains conjectural. One alternative is described by the superellipse equation (SE), a generalized version of EE, i.e., | x/a | n +| y/b | n = 1. To test the efficacy of EE versus SE to describe leaf geometry, the leaf shapes of two Michelia species (i.e., M. cavaleriei var. platypetala , and M. maudiae ), were investigated using 60 leaves from each species. Analysis shows that the majority of leaves (118 out of 120) had adjusted root-mean-square errors of < 0.05 for the nonlinear fitting of SE to leaf geometry, i.e., the mean absolute deviation from the polar point to leaf marginal points was smaller than 5% of the radius of a hypothesized circle with its area equaling leaf area. The estimates of n for the two species were ˂ 2, indicating that all sampled leaves conformed to SE and not to EE. This study confirms the existence of SE in leaves, linking this to its potential functional advantages, particularly the possible influence of leaf shape on hydraulic conductance.
EDITORIAL article Front. Plant Sci., 21 March 2023Sec. Functional Plant Ecology Volume 14 - 2023 | https://doi.org/10.3389/fpls.2023.1169558
Electroporation technique is widely used in biotechnology and medicine for the transport of various molecules through the membranes of biological cells. Different mathematical models of electroporation have been proposed in the literature to study pore formation in plasma and nuclear membranes. These studies are mainly based on models using a single isolated cell with a canonical shape. In this work, a space–time (x,y,t) multiphysics model based on quasi-static Maxwell’s equations and nonlinear Smoluchowski’s equation has been developed to investigate the electroporation phenomenon induced by pulsed electric field in multicellular systems having irregularly shape. The dielectric dispersion of the cell compartments such as nuclear and plasmatic membranes, cytoplasm, nucleoplasm and external medium have been incorporated into the numerical algorithm, too. Moreover, the irregular cell shapes have been modeled by using the Gielis transformations.
Abstract Egg geometry can be described using Preston's equation, which has seldom been used to calculate egg volume ( V ) and surface area ( S ) to explore S versus V scaling relationships. Herein, we provide an explicit re‐expression of Preston's equation (designated as EPE) to calculate V and S , assuming that an egg is a solid of revolution. The side (longitudinal) profiles of 2221 eggs of six avian species were digitized, and the EPE was used to describe each egg profile. The volumes of 486 eggs from two avian species predicted by the EPE were compared with those obtained using water displacement in graduated cylinders. There was no significant difference in V using the two methods, which verified the utility of the EPE and the hypothesis that eggs are solids of revolution. The data also indicated that V is proportional to the product of egg length ( L ) and maximum width ( W ) squared. A 2/3‐power scaling relationship between S and V for each species was observed, that is, S is proportional to ( LW 2 ) 2/3 . These results can be extended to describe the shapes of the eggs of other species to study the evolution of avian (and perhaps reptilian) eggs.
Measuring the inequality of leaf area distribution per plant (ILAD) can provide a useful tool for quantifying the influences of intra- and interspecific competition, foraging behavior of herbivores, and environmental stress on plants’ above-ground architectural structures and survival strategies. Despite its importance, there has been limited research on this issue. This paper aims to fill this gap by comparing four inequality indices to measure ILAD, using indices for quantifying household income that are commonly used in economics, including the Gini index (which is based on the Lorenz curve), the coefficient of variation, the Theil index, and the mean log deviation index. We measured the area of all leaves for 240 individual plants of the species Shibataea chinensis Nakai, a drought-tolerant landscape plant found in southern China. A three-parameter performance equation was fitted to observations of the cumulative proportion of leaf area vs. the cumulative proportion of leaves per plant to calculate the Gini index for each individual specimen of S. chinensis. The performance equation was demonstrated to be valid in describing the rotated and right shifted Lorenz curve, given that >96% of root-mean-square error values were smaller than 0.004 for 240 individual plants. By examining the correlation between any of the six possible pairs of indices among the Gini index, the coefficient of variation, the Theil index, and the mean log deviation index, the data show that these indices are closely related and can be used interchangeably to quantify ILAD.
Abstract Total leaf area per plant is an important measure of the photosynthetic capacity of an individual plant that together with plant density drives the canopy leaf area index, that is, the total leaf area per unit ground area. Because the total number of leaves per plant (or per shoot) varies among conspecifics and among mixed species communities, this variation can affect the total leaf area per plant and per canopy but has been little studied. Previous studies have shown a strong linear relationship between the total leaf area per plant (or per shoot) ( A T ) and the total number of leaves per plant (or per shoot) ( N T ) on a log–log scale for several growth forms. However, little is known whether such a scaling relationship also holds true for bamboos, which are a group of Poaceae plants with great ecological and economic importance in tropical, subtropical, and warm temperate regions. To test whether the scaling relationship holds true in bamboos, two dwarf bamboo species ( Shibataea chinensis Nakai and Sasaella kongosanensis ‘Aureostriatus’) with a limited but large number of leaves per culm were examined. For the two species, the leaves from 480 and 500 culms, respectively, were sampled and A T was calculated by summing the areas of individual leaves per culm. Linear regression and correlation analyses reconfirmed that there was a significant log–log linear relationship between A T and N T for each species. For S. chinensis , the exponent of the A T versus N T scaling relationship was greater than unity, whereas that of S. kongosanensis ‘Aureostriatus’ was smaller than unity. The coefficient of variation in individual leaf area increased with increasing N T for each species. The data reconfirm that there is a strong positive power‐law relationship between A T and N T for each of the two species, which may reflect adaptations of plants in response to intra‐ and inter‐specific competition for light.
Superquadrics are important models for part level-description in computer graphics and computer vision. Their power resides in their compact characterization. To further extend the representational power of superquadrics several methods have been proposed for local and global deformations. This notwithstanding, it is very difficult, for example, to represent polygons or polyhedrons using classical superquadrics. In this paper we present a new approach to model natural and abstract shapes for computer graphics, using a Generalized Superellipse Equation, which solves the problem of symmetries. Our approach provides an elegant analytical way to fold or unfold the coordinate axis systems like a fan, thereby generalizing superquadrics and superellipses (and hyperspheres in general) to supershapes for any symmetry, rational or irrational. Very compact representations of various shapes with different symmetries are possible and this provides opportunities for CAD at the level of graphics kernels, CAD-users and their clients. For example, parts and assemblies can be represented in very small file sizes allowing to use the 3-D solid model throughout the design and manufacturing process. Our approach presents an elegant way to use 3-D models both for solid modeling and boundary representations, for rigid as well as soft models.
Bamboos are among the economically most important plants world-wide. In Europe bamboos are used as ornamentals for gardens, but there is increasing interest for uses in ecological applications and as energy crops. Biotechnological techniques, inclu-ding tissue culture, in vitro hybridisation, molecular markers and genetic transformation are crucial for the future of bamboo. Micropropagation of bamboos has allowed to develop a new type of ornamental bamboo that can be produced yearround with a high quality / price ratio and distributed far more widely than classically propagated ornamental bamboos. Molecular markers are used in quality control procedures. Flowering of bamboos is still one of the greatest mysteries in botany, and breeding systems are non-existent. However, flowering can be induced reproducibly in tissue culture, both in seedlings and in adult bamboos, providing the only method for hybridisation. The flowering structures that are used are pseudospikelets, morphological features unique to the subfamily of Bambusoideae. These special propagules can be used for propagation, long term storage, for hybridisation and for genetic transformation. While flowering can be induced, controlled and reversed in tissue culture, a more fundamental approach to unravel the mechanisms of flowering include studies of cell division patterns and profiles of volatile components. Some applications of biotechnology for bamboo are presented, with emphasis on research strategies in a SME. Furthermore, all the techniques developed are of use not only in horticulture, but also in agriculture and forestry worldwide.
In this work, the electroporation phenomenon induced by pulsed electric field on different nucleated biological cells is studied. A nonlinear, non‐local, dispersive, and space–time multiphysics model based on Maxwell’s and asymptotic Smoluchowski’s equations has been developed to calculate the transmembrane voltage and pore density on both plasma and nuclear membrane perimeters. The irregular cell shape has been modeled by incorporating in the numerical algorithm the analytical functions pertaining to Gielis curves. The dielectric dispersion of the cell media has been modeled considering the multi‐relaxation Debye‐based relationship. Two different irregular nucleated cells have been investigated and their response has been studied applying both the dispersive and non‐dispersive models. By a comparison of the obtained results, differences can be highlighted confirming the need to make use of the dispersive model to effectively investigate the cell response in terms of transmembrane voltages, pore densities, and electroporation opening angle, especially when irregular cell shapes and short electric pulses are considered. Bioelectromagnetics. 2019;40:331–342. © 2019 Wiley Periodicals, Inc.
Plants have diverse leaf shapes that have evolved to adapt to the environments they have experienced over their evolutionary history. Leaf shape and leaf size can greatly influence the growth rate, competitive ability, and productivity of plants. However, researchers have long struggled to decide how to properly quantify the complexity of leaf shape. Prior studies recommended the leaf roundness index (RI = 4πA/P2) or dissection index (DI = √ ⁄), where P is leaf perimeter and A is leaf area. However, these two indices merely measure the extent of the deviation of leaf shape from a circle, which is usually invalid as leaves are seldom circular. In this study, we proposed a simple measure, named the ellipticalness index (EI), for quantifying the complexity of leaf shape based on the hypothesis that the shape of any oval leaf can be regarded as a variation from a standard ellipse. 2220 leaves from nine species of Magnoliaceae were sampled to check the validity of the EI. We also tested the validity of the Montgomery equation (ME), which assumes a proportional relationship between leaf area and the product of leaf length and width, because the EI actually comes from the proportionality coefficient of the ME. We also compared the ME with five other models of leaf area. The ME was found to be the best model for calculating leaf area based on consideration of the trade-off between model fit vs. complexity, which strongly supported the robustness of the EI for describing oval leaf shape. The new index can account for both leaf shape and size, and we conclude that it is a promising method for quantifying and comparing oval leaf shapes across species in future studies.
Abstract Recently, a universal equation by Narushin, Romanov, and Griffin (hereafter, the NRGE) was proposed to describe the shape of avian eggs. While NRGE can simulate the shape of spherical, ellipsoidal, ovoidal, and pyriform eggs, its predictions were not tested against actual data. Here, we tested the validity of the NRGE by fitting actual data of egg shapes and compared this with the predictions of our simpler model for egg shape (hereafter, the SGE). The eggs of nine bird species were sampled for this purpose. NRGE was found to fit the empirical data of egg shape well, but it did not define the egg length axis (i.e., the rotational symmetric axis), which significantly affected the prediction accuracy. The egg length axis under the NRGE is defined as the maximum distance between two points on the scanned perimeter of the egg's shape. In contrast, the SGE fitted the empirical data better, and had a smaller root‐mean‐square error than the NRGE for each of the nine eggs. Based on its mathematical simplicity and goodness‐of‐fit, the SGE appears to be a reliable and useful model for describing egg shape.
Abstract The interior and exterior Robin problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so-called “superformula” introduced by J. Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica © is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. The computed results are found to be in good agreement with the theoretical findings on Fourier series expansion presented by L. Carleson.
Many natural radial symmetrical shapes (e.g., sea stars) follow the Gielis equation (GE) or its twin equation (TGE). A supertriangle (three triangles arranged around a central polygon) represents such a shape, but no study has tested whether natural shapes can be represented as/are supertriangles or whether the GE or TGE can describe their shape. We collected 100 pieces of Koelreuteria paniculata fruit, which have a supertriangular shape, extracted the boundary coordinates for their vertical projections, and then fitted them with the GE and TGE. The adjusted root mean square errors (RMSEadj) of the two equations were always less than 0.08, and >70% were less than 0.05. For 57/100 fruit projections, the GE had a lower RMSEadj than the TGE, although overall differences in the goodness of fit were non-significant. However, the TGE produces more symmetrical shapes than the GE as the two parameters controlling the extent of symmetry in it are approximately equal. This work demonstrates that natural supertriangles exist, validates the use of the GE and TGE to model their shapes, and suggests that different complex radially symmetrical shapes can be generated by the same equation, implying that different types of biological symmetry may result from the same biophysical mechanisms.
Stomata are essential for the exchange of water vapour and atmospheric gases between vascular plants and their external environments. The stomatal geometries of many plants appear to be elliptical. However, prior studies have not tested whether this is a mathematical reality, particularly since many natural shapes that appear to be ellipses are superellipses with greater or smaller edge curvature than predicted for an ellipse. Compared with the ellipse equation, the superellipse equation includes an additional parameter that allows generation of a larger range of shapes. We randomly selected 240 stomata from each of four Magnoliaceae species to test whether the stomatal geometries are superellipses or ellipses. The stomatal geometries for most stomata (943/960) were found to be described better using the superellipse equation. The traditional "elliptical stomata hypothesis" resulted in an underestimation of the area of stomata, whereas the superellipse equation accurately predicted stomatal area. This finding has important implications for the estimation of stomatal area in studies looking at stomatal shape, geometry, and function.
Abstract The number and composition of species in a community can be quantified with α-diversity indices, including species richness ( R ), Simpson’s index ( D ), and the Shannon–Wiener index ( $$H^\prime$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> ). In forest communities, there are large variations in tree size among species and individuals of the same species, which result in differences in ecological processes and ecosystem functions. However, tree size inequality (TSI) has been largely neglected in studies using the available diversity indices. The TSI in the diameter at breast height (DBH) data for each of 999 20 m × 20 m forest census quadrats was quantified using the Gini index (GI), a measure of the inequality of size distribution. The generalized performance equation was used to describe the rotated and right-shifted Lorenz curve of the cumulative proportion of DBH and the cumulative proportion of number of trees per quadrat. We also examined the relationships of α-diversity indices with the GI using correlation tests. The generalized performance equation effectively described the rotated and right-shifted Lorenz curve of DBH distributions, with most root-mean-square errors (990 out of 999 quadrats) being < 0.0030. There were significant positive correlations between each of three α-diversity indices (i.e., R , D , and $$H^\prime$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> ) and the GI. Nevertheless, the total abundance of trees in each quadrat did not significantly influence the GI. This means that the TSI increased with increasing species diversity. Thus, two new indices are proposed that can balance α-diversity against the extent of TSI in the community: (1 − GI) × D , and (1 − GI) × $$H^\prime$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . These new indices were significantly correlated with the original D and $$H^\prime$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> , and did not increase the extent of variation within each group of indices. This study presents a useful tool for quantifying both species diversity and the variation in tree sizes in forest communities, especially in the face of cumulative species loss under global climate change.
From about 1993 to the present (1998), millions of plants of Fargesia murieliae, an ornamental bamboo, were flowering monocarpically in western Europe. All plants being ramets of a single genet, introduced in Europe about 80 years ago, the simultaneous flowering of all these ramets constitutes a single giant compound inflorescence. Flowering plants were subjected to different conditions of light intensity and temperature yielding different modes of development of flowering. Under low light intensity and high ambient temperatures (22-250C), reversion offlowering is ultimate- ly observed by development of vegetative shoots from basal buds offlowering culms. Under conditions that induce oxidative stress (full sun and drought) flowering and senescence proceed rapidly. The process of senescence is linked to oxidative stress as measured in elevated activities of superoxide dismutase and ascorbate peroxidase in the semelauctant inflorescences. By strong photoreduction of catalase activity, high light intensity is directly responsible for inactivation of one of the major defence mechanisms of the cells, thereby effectively accelerating flowering and senescence. In blades of spathes subtending the semelauctant inflorescences (short paracladial zone), DNA replication as well as loss of DNA from nuclei is observed. As this phenomenon is observed in green spathes prior to the development of the terminal inflorescence, DNA replication is a very early marker for senescence. A role for extra DNA synthesis prior to senescence is suggested in cytokinin biosynthesis.
A novel class of supershaped dielectric lens antennas, whose geometry is described by the three-dimensional extension of Gielis' formula, is introduced and analyzed. To this end, a hybrid approach based on geometrical and physical optics is adopted in order to properly model the multiple wave reflections occurring within the lens and the relevant impact on the radiation properties of the antenna under analysis. The developed modeling procedure has been validated by comparison with numerical results already reported in the scientific literature and, afterwards, applied to the electromagnetic characterization of a Gielis lens antenna with shaped radiation pattern.
Many geometries of plant organs can be described by the Gielis equation, a polar coordinate equation extended from the superellipse equation, r=a|cosm4φ|n2+|1ksinm4φ|n3−1/n1. Here, r is the polar radius corresponding to the polar angle φ; m is a positive integer that determines the number of angles of the Gielis curve when φ ∈ [0 to 2π); and the rest of the symbols are parameters to be estimated. The pentagonal radial symmetry of calyxes and corolla tubes in top view is a common feature in the flowers of many eudicots. However, prior studies have not tested whether the Gielis equation can depict the shapes of corolla tubes. We sampled randomly 366 flowers of Vinca major L., among which 360 had five petals and pentagonal corolla tubes, and six had four petals and quadrangular corolla tubes. We extracted the planar coordinates of the outer rims of corolla tubes (in top view) (ORCTs), and then fitted the data with two simplified versions of the Gielis equation with k = 1 and m = 5: r=acos54φn2+sin54φn3−1/n1 (Model 1), and r=acos54φn2+sin54φn2−1/n1 (Model 2). The adjusted root mean square error (RMSEadj) was used to evaluate the goodness of fit of each model. In addition, to test whether ORCTs are radially symmetrical, we correlated the estimates of n2 and n3 in Model 1 on a log-log scale. The results validated the two simplified Gielis equations. The RMSEadj values for all corolla tubes were smaller than 0.05 for both models. The numerical values of n2 and n3 were demonstrated to be statistically equal based on the regression analysis, which suggested that the ORCTs of V. major are radially symmetrical. It suggests that Model 1 can be replaced by the simpler Model 2 for fitting the ORCT in this species. This work indicates that the pentagonal or quadrangular corolla tubes (in top view) can both be modeled by the Gielis equation and demonstrates that the pentagonal or quadrangular corolla tubes of plants tend to form radial symmetrical geometries during their development and growth.
Preston’s equation is a general model describing the egg shape of birds. The parameters of Preston’s equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg’s actual length to two. This method assumes that the straight line through the two points on an egg’s profile separated by the maximum distance (i.e., the longest axis of an egg’s profile) is the mid-line. It hypothesizes that the photographed egg’s profile is perfectly bilaterally symmetrical, which seldom holds true because of photographic errors and placement errors. The existing parameter estimation method for Preston’s equation considers an angle of deviation for the longest axis of an egg’s profile from the mid-line, which decreases prediction errors to a certain degree. Nevertheless, this method cannot provide an accurate estimate of the coordinates of the egg’s center, and it leads to sub-optimal parameter estimation. Thus, it is better to account for the possible asymmetry between the two sides of an egg’s profile along its mid-line when fitting egg-shape data. In this paper, we propose a method based on the optimization algorithm (optimPE) to fit egg-shape data and better estimate the parameters of Preston’s equation by automatically searching for the optimal mid-line of an egg’s profile and testing its validity using profiles of 59 bird eggs spanning a wide range of existing egg shapes. We further compared this method with the existing one based on multiple linear regression (lmPE). This study demonstrated the ability of the optimPE method to estimate numerical values of the parameters of Preston’s equation and provide the theoretical egg length (i.e., the distance between two ends of the mid-line of an egg’s profile) and the egg’s maximum breadth. This provides a valuable approach for comparing egg shapes among conspecifics or across different species, or even different classes (e.g., birds and reptiles), in future investigations.
A uniform description of natural shapes and phenomena is an important goal in science.Such description should check some basic principles, related to 1) the complexity of the model, 2) how well its its real objects, phenomena and data, and 3) a direct connection with optimization principles and the calculus of variations.In this article, we present nine principles, three for each group, and we compare some models with a claim to universality.It is also shown that Gielis Transformations and power laws have a common origin in conic sections.
The Lamé curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lamé curve in the polar coordinate system by introducing additional parameters (n1, n2, n3; m): rφ=1Acosm4φn2+1Bsinm4φn3−1/n1, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar angle φ; A, B, n1, n2 and n3 are parameters to be estimated; m is the positive real number that determines the number of angles of the Gielis curve. Most prior studies on the Gielis equation focused mainly on its applications. However, the Gielis equation can also generate a large number of shapes that are rotationally symmetric and axisymmetric when A = B and n2 = n3, interrelated with the parameter m, with the parameters n1 and n2 determining the shapes of the curves. In this paper, we prove the relationship between m and the rotational symmetry and axial symmetry of the Gielis curve from a theoretical point of view with the condition A = B, n2 = n3. We also set n1 and n2 to take negative real numbers rather than only taking positive real numbers, then classify the curves based on extremal properties of r(φ) at φ = 0, π/m when n1 and n2 are in different intervals, and analyze how n1, n2 precisely affect the shapes of Gielis curves.
Although many fruit geometries resemble a solid of revolution, this assumption has rarely been rigorously examined. To test this assumption, 574 fruits of Canarium album (Lour.) DC. which appear to have an ellipsoidal shape, were examined to determine the validity of a general avian-based egg-shape equation, referred to as the explicit Preston equation (EPE). The assumption that the C. album fruit geometry is a solid of revolution is tested by applying the volume formula for a solid of revolution using the EPE. The goodness of fit of the EPE was assessed using the adjusted root-mean-square error (RMSEadj). The relationship between the observed volume (Vobs) of each fruit, as measured by water displacement in a graduated cylinder, and the predicted volumes (Vpre) based on the EPE was also evaluated using the equation Vpre = slope * Vobs. All the RMSEadj values were smaller than 0.05, which demonstrated the validity of the EPE based on C. album fruit profiles. The 95% confidence interval of the slope of Vpre vs. Vobs included 1.0, indicating that there was no significant difference between Vpre and Vobs. The data confirm that C. album fruits are solids of revolution. This study provides a new approach for calculating the volume and surface area of geometrically similar fruits, which can be extended to other species with similar fruit geometries to further explore the ontogeny and evolution of angiosperm reproductive organs.
The interior and exterior Dirichlet problems for the Laplace equation in k-type Gielis domains are analytically addressed by using a suitable Fourier-like technique. A dedicated numerical procedure based on the computer-aided algebra
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Abstract A Super‐Formula‐based compact ultra‐wideband (UWB) printed monopole antenna is proposed in this paper. The presented design is implemented by using inexpensive FR‐4 laminates, and is optimized so as to enable excellent radio frequency performance in terms of impedance matching, realized gain, total efficiency, radiation pattern characteristics, maximal group delay variation, and system fidelity factor (SFF), while covering the Federal Communications Commission UWB channel 5, from 6.24 to 6.74 GHz, and channel 9, from 7.74 to 8.24 GHz. The proposed antenna solution has been characterized both passively and actively using a commercially available indoor real‐time positioning system, which relies on the phase‐difference‐of‐arrival algorithm. The collected measurement results demonstrate the high accuracy in real‐time positioning achieved with the developed antenna both in terms of distance and angle.
Superquadrics are important models for part level-description in computer graphics and computer vision. Their power resides in their compact characterization. To further extend the representational power of superquadrics several methods have been proposed for local and global deformations. This notwithstanding, it is very difficult, for example, to represent polygons or polyhedrons using classical superquadrics. In this paper we present a new approach to model natural and abstract shapes for computer graphics, using a Generalized Superellipse Equation, which solves the problem of symmetries. Our approach provides an elegant analytical way to fold or unfold the coordinate axis systems like a fan, thereby generalizing superquadrics and superellipses (and hyperspheres in general) to supershapes for any symmetry, rational or irrational. Very compact representations of various shapes with different symmetries are possible and this provides opportunities for CAD at the level of graphics kernels, CAD-users and their clients. For example, parts and assemblies can be represented in very small file sizes allowing to use the 3-D solid model throughout the design and manufacturing process. Our approach presents an elegant way to use 3-D models both for solid modeling and boundary representations, for rigid as well as soft models.
Functional plant traits include a plant’s phenotypic morphology, nutrient element characteristics, and physiological and biochemical features, reflecting the survival strategies of plants in response to environmental changes [...]
Synopsis The proportions in the size of the avian egg albumen, yolk, and shell are crucial for understanding bird survival and reproductive success because their relationships with volume and surface area can affect ecological and life history strategies. Prior studies have focused on the relationship between the albumen and the yolk, but little is known about the scaling relationship between eggshell mass and shape and the mass of the albumen and the yolk. Toward this end, 691 eggs of six precocial species were examined, and their 2-D egg profiles were photographed and digitized. The explicit Preston equation, which assumes bilateral symmetrical geometry, was used to fit the 2-D egg profiles and to calculate surface areas and volumes based on the hypothesis that eggs can be treated as solids of profile revolution. The scaling relationships of eggshell mass (Ms), albumen mass (Ma), and yolk mass (My), as well as the surface area (S), volume (V), and total mass (Mt) were determined. The explicit Preston equation was validated in describing the 2-D egg profiles. The scaling exponents of Ma vs. Ms, My vs. Ms, and My vs. Ma were smaller than unity, indicating that increases in Ma and My fail to keep pace with increases in Ms, and that increases in My fail to keep pace with increases in Ma. Therefore, increases in unit nutrient contents (i.e., the yolk) involve disproportionately larger increases in eggshell mass and disproportionately larger increases in albumen mass. The data also revealed a 2/3-power scaling relationship between S and V for each species, that is, the simple Euclidean geometry is obeyed. These findings help to inform our understanding of avian egg construction and reveal evolutionary interspecific trends in the scaling of egg shape, volume, mass, and mass allocation.
In the present paper we consider the "bulky knots'' and "bulky links'', which appear after cutting a Generalized Möbius Listing's GML_3^n body (whose radial cross section is a plane 3 -symmetric figure with three vertices) along different Generalized Möbius Listing's surfaces GML_2^n situated in it. This article is aimed to investigate the number and geometric structure of the independent objects appearing after such a cutting process of GML_3^n bodies. In most cases we are able to count the indices of the resulting mathematical objects according to the known tabulation for Knots and Links of small complexity.
In this communication, the use of Gielis transformation to design more compact metamaterial unit cells is explored. For this purpose, transformed complementary split ring resonators and spiral resonators are coupled to micro-strip lines and theirbehaviour is investigated. The obtained results confirm that the useof the considered class of supershaped geometries enables the synthesis of very compact scalable microwave components.
We generalize the Gielis Superformula by extending the R. Chacón approach, but avoiding the use of Jacobi elliptic functions. The obtained results are extended to the three-dimensional case. Several new shapes are derived by using the computer algebra system Mathematica .
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Abstract The Robin problem for the Helmholtz equation in normal-polar annuli is addressed by using a suitable Fourier-Hankel series technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica © is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
The chapter will describe the potential of the swarm intelligence and in particular quantum PSO-based algorithm, to solve complicated electromagnetic problems. This task is accomplished through addressing the design and analysis challenges of some key real-world problems. A detailed definition of the conventional PSO and its quantum-inspired version are presented and compared in terms of accuracy and computational burden. Some theoretical discussions concerning the convergence issues and a sensitivity analysis on the parameters influencing the stochastic process are reported.
Electroporation is a non-thermal electromagnetic phenomenon widely used in medical diseases treatment. Different mathematical models of electroporation have been proposed in literature to study pore evolution in biological membranes. This paper presents a nonlinear dispersive multiphysic model of electroporation in irregular shaped biological cells in which the spatial and temporal evolution of the pores size is taken into account. The model solves Maxwell and asymptotic Smoluchowski equations and it describes the dielectric dispersion of cell media using a Debye-based relationship. Furthermore, the irregular cell shape has been modeled using the Gielis superformula. Taking into account the cell in mitosis phase, the electroporation process has been studied comparing the numerical results pertaining the model with variable pore radius with those in which the pore radius is supposed constant. The numerical analysis has been performed exposing the biological cell to a rectangular electric pulse having duration of 10\\ \\mu\\mathrm{s}. The obtained numerical results highlight considerable differences between the two different models underling the need to include into the numerical algorithm the differential equation modeling the spatial and time evolution of the pores size.
Gielis curves and surfaces can describe a wide range of natural shapes and they have been used in various studies in biology and physics as descriptive tool.This has stimulated the generalization of widely used computational methods.Here we show that proper normalization of the Levenberg-Marquardt algorithm allows for efficient and robust reconstruction of Gielis curves, including self-intersecting and asymmetric curves, without increasing the overall complexity of the algorithm.Then, we show how complex curves of k-type can be constructed and how solutions to the Dirichlet problem for the Laplace equation on these complex domains can be derived using a semi-Fourier method.In all three methods, descriptive and computational power and efficiency is obtained in a surprisingly simple way.
The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.
Supershapes are used in Parametric Design to model, literally, thou-sands of natural and man-made shapes with a single 6 parameter formula. However, users are left to probe such a rich yet dense collection of supershapes using a set of independent 1-D sliders. Some of the formula's parameters are non-linear in nature, making them particularly difficult to grasp with conventional 1-D sliders alone. VR appears as a promising setting for Parametric Design with supershapes since it empowers users with more natural visual inspection and shape browsing techniques, with multiple solutions being displayed at once and the possibility to design more interesting forms of slider interaction. In this work, we propose VR shape widgets that allow users to probe and select supershapes from a multitude of solutions. Our designs take leverage on thumbnails, mini-maps, haptic feedback and spatial interaction, while supporting 1-D, 2-D and 3-D supershape parameter spaces. We conducted a user study (N = 18) and found that VR shape widgets are effective, more efficient, and natural than conventional VR 1-D sliders while also usable for users without prior knowledge on supershapes. We also found that the proposed VR widgets provide a quick overview of the main supershapes, and users can easily reach the desired solution without having to perform fine-grain handle manipulations.
Bamboo is an important component in subtropical and tropical forest communities. The plant has characteristic long lanceolate leaves with parallel venation. Prior studies have shown that the leaf shapes of this plant group can be well described by a simplified version (referred to as SGE-1) of the Gielis equation, a polar coordinate equation extended from the superellipse equation. SGE-1 with only two model parameters is less complex than the original Gielis equation with six parameters. Previous studies have seldom tested whether other simplified versions of the Gielis equation are superior to SGE-1 in fitting empirical leaf shape data. In the present study, we compared a three-parameter Gielis equation (referred to as SGE-2) with the two-parameter SGE-1 using the leaf boundary coordinate data of six bamboo species within the same genus that have representative long lanceolate leaves, with >300 leaves for each species. We sampled 2000 data points at approximately equidistant locations on the boundary of each leaf, and estimated the parameters for the two models. The root–mean–square error (RMSE) between the observed and predicted radii from the polar point to data points on the boundary of each leaf was used as a measure of the model goodness of fit, and the mean percent error between the RMSEs from fitting SGE-1 and SGE-2 was used to examine whether the introduction of an additional parameter in SGE-1 remarkably improves the model’s fitting. We found that the RMSE value of SGE-2 was always smaller than that of SGE-1. The mean percent errors among the two models ranged from 7.5% to 20% across the six species. These results indicate that SGE-2 is superior to SGE-1 and should be used in fitting leaf shapes. We argue that the results of the current study can be potentially extended to other lanceolate leaf shapes.
Aim of this chapter is analytical representation of one wide class of geometric figures (lines, surfaces and bodies) and their complicated displacements. The accurate estimation of physical characteristics (such as volume, surface area, length, or other specific parameters) relevant to human organs is of fundamental importance in medicine. One central idea of this article is, in this respect, to provide a general methodology for the evaluation, as a function of time, of the volume and center of gravity featured by moving of one class of bodies used of describe different human organs.
This research study reports the assessment of complementary split ring resonators based on Gielis transformation as basic elements for the design of high-performance microwave components in printed technology. From the electromagnetic simulation of said structures, suitable equivalent circuit models are extracted and analyzed. Physical prototypes are fabricated and tested for design validation. The obtained results confirm that the adoption of supershaped geometries enables the synthesis of very compact scalable microwave filters.
In nature, the two-dimensional (2D) profiles of fruits from many plants often resemble ellipses. However, it remains unclear whether these profiles strictly adhere to the ellipse equation, as many natural shapes resembling ellipses are actually better described as superellipses. The superellipse equation, which includes an additional parameter n compared to the ellipse equation, can generate a broader range of shapes, with the ellipse being just a special case of the superellipse. To investigate whether the 2D profiles of fruits are better described by ellipses or superellipses, we collected a total of 751 mature and undamaged fruits from 31 naturally growing plants of Cucumis melo L. var. agrestis Naud. Our analysis revealed that most adjusted root-mean-square errors (> 92% of the 751 fruits) for fitting the superellipse equation to the fruit profiles were consistently less than 0.0165. Furthermore, there were 638 of the 751 fruits (ca. 85%) with the 95% confidence intervals of the estimated parameter n in the superellipse equation not including 2. These findings suggest that the profiles of C. melo var. agrestis fruits align more closely with the superellipse equation than with the ellipse equation. This study provides evidence for the existence of the superellipse in fruit profiles, which has significant implications for studying fruit geometries and estimating fruit volumes using the solid of revolution formula. Furthermore, this discovery may contribute to a deeper understanding of the mechanisms driving the evolution of fruit shapes.
While there are well-known synthetic methods in the literature for finding the image of a point under circular inversion in l2-normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in creating a synthetic construction of the circular inversion in l1-normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l1-norm.
Building on previous work with generalized conic sections, in particular with the Superformula, we introduce ultraflexibility instead of rigidity as encoded in the geometry of Euclid and Descartes. By considering points as ultra-extensible primitives, Point-Manifolds are generated having shape, size and history, defining a very wide range of natural and abstract shapes.
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<p>The electromagnetic analysis of a special class of 3D dielectric lens antennas is described in detail. This new class of lens antennas has a geometrical shape defined by the three-dimensional extension of Gielis' formula. The analytical description of the lens shape allows the development of a dedicated semianalytical hybrid modeling approach based on geometrical tube tracing and physical optic. In order to increase the accuracy of the model, the multiple reflections occurring within the lens are also taken into account.</p>
In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, it is well known that the Möbius phenomenon, whereby the cutting process yields only one body, occurs only in even polygons with an even number of vertices and sides, and only in the specific when the knife cuts through the center of the polygon.This knife cuts from vertex to vertex, vertex to side or side to side, cutting exactly two points on the boundary of the polygon.This is called a chordal knife, in connection to the chord cutting a circle.If the knife is a radial knife, i.e. it cuts only one point of the boundary, the Möbius phenomenon can occur both in odd and even polygons, but only when the radial knife cuts the center of the polygon.One finding is the reduction of a problem in 3D (with internal geometry) to a planar problem and the concomitant reduction of the analytic representation with multiple parameters to a few only.The shape of the cross section and number of twisting in the 3D representation suffice and reduce the problem to cutting of regular polygons and cyclic permutations.
The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical har-monic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are de-fined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained
One of the defining characteristics of Greek legacy was rational thinking, and from that sprang forward the notion that mathematics has an intimate relationship to the workings of the world. According to Pythagoras, Everything is Number (and for Pythagoras geometry and number were inextractably linked). According to Richard Feynman (1918–1988): "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form" [69]. Fortunately, simple rules are a basic feature of this language, by whatever name they are known, addition and multiplication, means, cubes or the monomiality principle.
In this work, a nonlinear dispersive multiphysic model based on Maxwell and asymptotic Smoluchowsky equations has been developed to analyze the electroporation phenomenon induced by pulsed electric field on biological cells. The irregular plasma membrane geometry has been modeled by incorporating in the numerical algorithm the Gielis superformula as well as the dielectric dispersion of the plasma membrane has been modeled using the multi-relaxation Debye-based relationship. The study has been carried out with the aim to compare our model implementing a thin plasma membrane with the simplified model in which the plasma membrane is modeled as a distributed impedance boundary condition. The numerical analysis has been performed exposing the cell to external electric pulses having rectangular shapes. By an inspection of the obtained results, significant differences can be highlighted between the two models confirming the need to incorporate the effective thin membrane into the numerical algorithm to well predict the cell response to the pulsed electric fields in terms of transmembrane voltages and pore densities, especially when the cell is exposed to external nanosecond pulses.
Optimization is a widely used concept in many fields, such as engineering, economics, management, physical sciences, and social sciences.Its purpose is to identify the global maximum or minimum of a fitness function.Finding all optimal points of an objective function can aid in selecting a robust design that simultaneously considers various constraints and performance criteria.Designers of microwave and antenna systems face the challenge of finding optimal solutions for electromagnetic problems of increasing complexity.This can be a difficult task as it involves evaluating electromagnetic fields in three dimensions, considering a large number of parameters and complex constraints, and dealing with nondifferentiable and discontinuous regions.These optimization problems are often non-linear and more challenging to solve than linear ones, especially when many locally optimal solutions are in the feasible region.When developing electromagnetic systems, it is essential to carefully consider how the different design elements interact with each other.Instead of relying on brute-force computational techniques, experts use advanced optimization procedures to achieve the best results.These procedures can be grouped into two categories:Electromagnetic design problems involve optimizing multiple parameters that are nonlinearly related to objective functions.Traditional optimization techniques require significant computational resources that grow exponentially as the problem size increases.Therefore, a method that can produce good results with moderate memory and computational resources is desirable.Bioinspired optimization methods, such as particle swarm optimization (PSO), are known for their computational efficiency and are commonly used in various scientific and technological fields.In this article we explore the potential of advanced PSO-based algorithms to tackle challenging electromagnetic design and analysis problems faced in real-life applications.It provides a detailed comparison between conventional PSO and its quantum-inspired version regarding accuracy and computational costs.Additionally, theoretical insights on convergence issues and sensitivity analysis on parameters influencing the stochastic process are reported.The utilization of a novel quantum PSO-based algorithm in advanced scenarios, such as reconfigurable and shaped lens antenna synthesis, is illustrated.The hybrid modeling approach, based on the unified geometrical description enabled by the Gielis Transformation, is applied in combination with a suitable quantum PSO-based algorithm, along with a geometrical tube tracing and physical optics technique for solving the inverse problem aimed at identifying the geometrical parameters that yield optimal antenna performance.
Generalized Möbius-Listing bodies and surfaces are generalizations of the classic Möbius band. The original motivation is that for solutions of boundary value problems the knowledge of the domain is essential. In previous papers cutting of GML bodies with cross section symmetrical disks with symmetry 2, 3, 4, 5 and 6 have been classified. In this paper we solve the general case, using regular m-gons as cross section. The 3D problem is reduced to the problem of cutting regular m-polygons with d-knives, related to the number of divisors of m. The problem has both a geometrical and topological solution, and has many connections to other fields of mathematics.
The interaction of diatoms with sunlight is fundamental in order to deeply understand their role in terrestrial ecology and biogeochemistry, essentially due to their massive contribution to global primary production through photosynthesis and its effect on carbon, oxygen and silicon cycles. Following the journey of light through natural waters, its propagation through the intricate frustule micro-and nano-structure and, finally, its fate inside the photosynthetic machinery of the living cell requires several mathematical and computational models in order to accurately describe all the involved phenomena taking place at different space scales and physical regimes. In this chapter, we review the main analytical models describing the underwater optical field, the essential numerical algorithms for the study of photonic properties of the diatom frustule seen as a natural metamaterial, as well as the principal models describing photon harvesting in diatom plastids and methods for complex EM propagation problems and wave propagation in dispersive materials with multiple relaxation times. These mathematical methods will be integrated in a unifying geometric perspective.
Entropy is originally a concept from thermodynamics that distinguishes between useful and useless energy 1 .This led to the Second Law of Thermodynamics, which states that entropy always increases.The Hamiltonian principle states that the differential equation for the Second Law is equivalent to the integral equation for Least Action.In the information age, entropy has been found to be related to complexity.There are two main approaches, one of which is based on C. Shannon (1916Shannon ( -2001) ) and the other on A. N. Kolmogorov (1903N.Kolmogorov ( -1987)).The former is communication theory, while the latter is the basis of Algorithmic Information Theory (AIT), which studies the shortest algorithm for encoding a message that yields the "best possible compression".This is of course also usage-based and refers to 'optimal' methods of data processing.The commonly used example is a particular sequence of binary digits, symbols or characters.Each of these sequences is called a message.Two sequences (or strings) are:1 "Useful energy" is somewhat anthropocentric, in the sense of "usefulness for humans".Our scientific and technological worldviews are largely dominated by the concepts of entropy and complexity.Originating in 19th-century thermodynamics, the concept of entropy merged with information in the last century, leading to definitions of entropy and complexity by Kolmogorov, Shannon and others.In its simplest form, this worldview is an application of the normal rules of arithmetic.In this worldview, when tossing a coin, a million heads or tails in a row is theoretically possible, but impossible in practice and in real life.On this basis, the impossible (in the binary case, the outermost entries of Pascal's triangle and for large values of ) can be safely neglected, and one can concentrate fully on what is common and what conforms to the law of large numbers, in fields ranging from physics to sociology and everything in between.However, in recent decades it has been shown that what is most improbable tends to be the rule in nature.Indeed, if one combines the outermost entries and with the normal rules of arithmetic, either addition or multiplication, one obtains Lamé curves and power laws respectively.In this article, some of these correspondences are highlighted, leading to a double conclusion.First, Gabriel Lamé's geometric footprint in mathematics and the sciences is enormous.Second, conic sections are at the core once more.Whereas mathematics so far has been exclusively the language of patterns in the sciences, the door is opened for mathematics to also become the language of the individual.The probabilistic worldview and Lamé's footprint can be seen as dual methods.In this context, it is to be expected that the notions of information, complexity, simplicity and redundancy benefit from this different viewpoint.
One of the most beautiful stories in botany is that of the Orchid Angraecum sesquipedale.
Understanding life is one of the major challenges for science in the 21st century. Despite the exponentially growing mountains of data in the life sciences, the challenge of developing geometrical models, always at the core in eras of scientific progress (Newton, Riemann, Einstein), remains completely open. Marcel Berger wrote explicitly: ''Present models of geometry, even if quite numerous, are not able to answer various essential questions. For example: among all possible configurations of a living organism, describe its trajectory (life) in time'' [6]. We are still far from describing life mathematically, despite the numerous successful applications of mathematics in the life sciences.
Plants are continuously communicating with an ever-changing environments where macro and micro-effects interact, having an effect on morphogenesis, making all leaves, all tree rings and all flowers a bit different from one another. This is why we explicitly opt for a geometrical model: our purpose is not to build nice visuals, our goals is to develop abstract plants. The parameters in the abstract plant enable us to measure the effect of interactions with the environment, which has to take into account natural curvature conditions [94]. Making nice and appealing virtual plants cannot be the goal of our endeavor. Indeed, introducing parameters and choosing values of them with the aim of obtaining a virtual plant resembling a given specimen contradicts our aim of using the "abstract" plant in the study of the morphogenesis as a time sequence of adaptations to internal and external stresses acting on the plants. The model constructed should be generic, i.e.; free of in-built constraints depending on results or theories contained in those to be tested in the actual research.
Relations between arithmetic and geometric means will take center stage, so I thought it not amiss to devote a complete chapter to point out the ancient roots of these notions and the relations between seemingly disparate fields.
Lamé curves are not some strange specimen in the field of mathematics. Considering them as pure numbers—the outermost values of any row of Pascal's Triangle—they are dual to the common "geometric mean" approach.
One uniform, Pythagorean-compact description with all the associated characteristics that come for free allows for the study of natural shapes in a natural way.
In Lamé Ovals, Gridgeman writes: "The set of Lamé curves for A = B and n > 1 fills the area between a square and its inscribed circle. A further generalization that immediately suggests itself is the coverage of the analogous curves that lie between other regular polygons and their inscribed circles. This is merely a remark" [92].
Curvature, in all of its aspects is at the core of geometry. Going straight is central to geometry, but it is also a fiction, since we live in a curved world with gravitational attraction in landscapes of mountains, valleys and planes. Nevertheless it is a very useful fiction, and the basis of Euclidean geometry and our science. How then to define being curved, intrinsically and/or as a deviation from planarity? This is one of the central questions in mathematics and geometry; we have derived Euclidean geometry, one geometry that is in accordance with our intuition and our position on earth (with its gravitational field).
A new class of irregularly shaped dielectric lens antennas with a supershaped microstrip antenna feeder is presented and detailed in this work. The surface of the lens antenna and the feeder shape have been modelled by using the three and two-dimensional Gielis formula, respectively. The antenna design has been carried out by integrating an home-made software tool with the CST Microwave Studio®. The radiation properties of the whole antenna system have been evaluated using a dedicated high-frequency technique based on the tube tracing approximation. Moreover, the effects due to the multiple internal reflections have been properly modeled. The proposed model was applied to study unusual and complex lens antenna systems with the aim to design special radiation characteristics.
As generalization of Lamé curves, the Gielis formula is deeply rooted in a long tradition in mathematics of developing compact methods for shape description, striving for a uniform description.
Möbius bands have been studied extensively, mainly in topology.Generalized Möbius-Listing surfaces and bodies providing a full geometrical generalization, is a quite new field, motivated originally by solutions of boundary value problems.Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces.In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems.However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably.Our research is motivated by this reduction of complexity.In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, the conditions under which a single body results, displaying the Möbius phenomenon of a one-sided body, have been determined for even and odd polygons.These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin.The Möbius phenomenon is important, since the process of cutting (or separation of zones in a GML body in general) then results in a single body, not in different, intertwined domains.In all previous works it was assumed that the cross section of the GML bodies is constant, but the main result of this paper is that it is sufficient that only one cross section on the whole GML structure meets the conditions for the Möbius phenomenon to occur.Several examples are given to illustrate this.
Our understanding of quantum phenomena often begins with simple particle-in-a-box style problems, the solutions of which introduce the student to foundational quantum concepts such as degeneracy and quantization. Simple model geometries of confinement afford analytic solutions, which are readily derivable, easily manipulable, and provide a unique sandbox of exploration accessible at the undergraduate level. In the current work, these model problems are explored in a variety of ways. Firstly, through a historical lens - orienting them to the birth and development of quantum physics. Then, via an organizing syntax. This framework allows the interested student to orient the diverse multidisciplinary literature that has evolved around these problems. Finally, through consideration of the shape element of the syntax, the superformula a simple extension of the equation describing a circle is introduced and discussed.
Generalized Möbius-Listing surfaces and bodies generalize Möbius bands, and this research was motivated originally by solutions of boundary value problems.Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces.In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems.However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably.These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin.
While there are well-known synthetic methods in the literature to find the image of a point under circular inversion in l2−normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in giving a synthetic construction for the circular inversion in l1−normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l1−norm.
The second issue of Volume 35 of the Symmetry: Culture and Science journal arrives shortly before the Symmetry Festival 2024, marking its debut in Italy, in the historic city of Pisa, from July 17 to July 20. This event will be a tribute to the founder of the International Symmetry Association and a friend to many of our readers, György Darvas, who passed away in late 2023. In Volume 35, Number 2, we celebrate Darvas’s ongoing legacy of symmetry as a unifying principle across diverse disciplines. This issue presents a diverse range of interdisciplinary research in mathematics, science, art, education, and architecture, emphasizing the interconnectedness inherent in symmetry. We hope these contributions inspire further exploration and appreciation of symmetry’s multifaceted nature.
Welcome to the first issue of Volume 35 for 2024 of Symmetry: Culture and Science. As we begin the second year of our editorial work, Johan and I are happy to continue offering insights into the world of symmetry. To endure the journal’s legacy and tradition, we investigate the different symmetry dimensions, from cultural models to mathematical concepts and their intersection within different fields. As we undertake this intellectual voyage through the multifaceted symmetry domain, we invite you to explore, ponder, and engage with the rich tapestry of ideas presented in this issue. We are also excited to announce the upcoming Symmetry Festival 2024, held in July in Pisa, where academics, scholars, researchers, and enthusiasts will converge to celebrate and explore the myriad facets of symmetry. We sincerely thank the authors, reviewers, and editorial team whose dedication and contributions have made this issue possible. We hope reading this issue will inspire you to walk in new ways into the territory never imagined of symmetry in all its forms.
This study addresses the critical need for efficient phenotyping methods in plant ecology by exploring predictive models for total leaf area per shoot ( A T ) and total leaf dry mass per shoot ( M T ), which are both key determinants of photosynthetic capacity and carbon allocation, using two fast-growing bamboo species ( Indocalamus decorus and I. longiauritus ) as proof of concept. Traditional approaches to measuring these traits are destructive and labor-intensive, motivating our exploration of non-destructive proxies based on one-dimensional leaf metrics. We validated the Montgomery equation for individual leaves, confirming a robust proportional relationship between leaf area ( A ) and the product of length and width ( LW ) in both Indocalamus species ( k ≈ 0.72). Extending this to the shoot level, the Montgomery-Koyama-Smith equation (MKSE) revealed significant proportionality between total leaf area ( A T ) and the composite metric L KS W KS (where L KS denotes the sum of leaf widths and W KS denotes maximum leaf length, and the subscript “KS” stands for Koyama-Smith). However, power-law scaling analysis demonstrated allometric, non-isometric relationships for A T vs. L KS W KS (with a scaling exponent α &lt; 1), indicating diminishing leaf area expansion per unit dimensional increase, and A T vs. total leaf dry mass ( M T ) (α &lt; 1), indicating an increased biomass investment per unit area (i.e., increasing leaf mass per unit area) in larger shoots. These findings validate using simplified one-dimensional metrics that enable accurate, non-destructive predictions of shoot-level functional traits, advancing phenotyping in bamboo ecology, which may hold true more generally for other types of plant species.
Building on earlier work with generalised conic sections, we use the superformula to introduce ultra-flexibility instead of rigidity as encoded in the geometry of Euclid and Descartes. By considering Points as ultra-extensible primitives, we define Points endowed with shape, size, and historical continuity. This Point-Theory of Morphogenesis addresses multiple challenges for a mathematical theory of morphogenesis for both natural and abstract shapes. The theory is formalised by a minimal set of one definition, two axioms, and two postulates.
https://sites.google.com/view/geometree-symposium/2025-edition/issbg-2025 The fourth edition of the International Symposium on Square B…
