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“Universal natural shapes : from unifying shape description to simple methods for shape analysis and boundary value problems”. Gielis J, Caratelli D, Fougerolle Y, Ricci PE, Tavkelidze I, Gerats T, PLoS ONE 7, e29324 (2012). http://doi.org/10.1371/JOURNAL.PONE.0029324
Abstract: 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1371/JOURNAL.PONE.0029324
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“Why does not the leaf weight-area allometry of bamboos follow the 3/2-power law?”.Lin S, Shao L, Hui C, Song Y, Reddy GVP, Gielis J, Li F, Ding Y, Wei Q, Shi P, Reddy GVP, Frontiers in plant science 9, 583 (2018). http://doi.org/10.3389/FPLS.2018.00583
Abstract: 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 (A), density (ρ), length (L), thickness (T), and weight (W). Because the edge of a bamboo leaf follows a simplified two-parameter Gielis equation, we could show that A ∝ L2 and that A ∝ T2. This then allowed us to derive the density-thickness allometry ρ ∝ Tb and the weight-area allometry W ∝ A(b+3)/2 ≈ A9/8, where b 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.3389/FPLS.2018.00583
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“A simple way to calculate the volume and surface area of avian eggs”. Shi P, Chen L, Quinn BK, Yu K, Miao Q, Guo X, Lian M, Gielis J, Niklas KJ, Annals of the New York Academy of Sciences 1524, 118 (2023). http://doi.org/10.1111/NYAS.15000
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 (LW2)(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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
Impact Factor: 5.2
DOI: 10.1111/NYAS.15000
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“Evidence that Chinese white olive (Canarium album(Lour.) DC.) fruits are solids of revolution”. Wang L, Shi P, Chen L, Gielis J, Niklas KJ, Botany letters , 1 (2023). http://doi.org/10.1080/23818107.2023.2238020
Abstract: 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
Impact Factor: 1.5
DOI: 10.1080/23818107.2023.2238020
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“Stomatal shape described by a superellipse in four Magnoliaceae species”. Li Q, Niklas KJJ, Niinemets U, Zhang L, Yu K, Gielis J, Gao J, Shi P, Botany letters , 1 (2023). http://doi.org/10.1080/23818107.2023.2234443
Abstract: 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
Impact Factor: 1.5
DOI: 10.1080/23818107.2023.2234443
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“Analysis and synthesis of supershaped dielectric lens antennas”. Bia P, Caratelli D, Mescia L, Gielis J, IET microwaves, antennas and propagation 9, 1497 (2015). http://doi.org/10.1049/IET-MAP.2015.0091
Abstract: 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.
Keywords: A1 Journal article; Engineering sciences. Technology; Mass communications; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1049/IET-MAP.2015.0091
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“On means, polynomials and special functions”. Gielis J, Verhulst R, Caratelli D, Ricci PE, Tavkhelidze I, The teaching of mathematics 17, 1 (2014)
Keywords: A1 Journal article; Educational sciences; Sustainable Energy, Air and Water Technology (DuEL)
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“Comparison of the performance of two polar equations in describing the geometries of elliptical fruits”. Yao W, Hui C, Wang L, Wang J, Gielis J, Shi P, Botany letters (2024). http://doi.org/10.1080/23818107.2024.2350014
Abstract: 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.
Keywords: A1 Journal article; Antwerp engineering, PhotoElectroChemistry & Sensing (A-PECS)
Impact Factor: 1.5
DOI: 10.1080/23818107.2024.2350014
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