Ellipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law

Mapping Intimacies ◽

10.7287/peerj.preprints.27421v2 ◽

2018 ◽

Author(s):

Kai Xu

Keyword(s):

Major Axis ◽

Theoretical Explanation ◽

Minor Effect ◽

Cell Area ◽

Semi Major Axis ◽

Polygonal Cell ◽

Cell Topology ◽

Form Deviation ◽

A Minor ◽

Ellipse Packing

Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear. Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis. Results: The average form deviation of vertexes from the fitted ellipse was 0 \(\pm\) 3.1 % (8,291 vertices in 1375 cells were examined). The area of polygonal cell was 0.9 \(\pm\)0.1 times of the area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells could be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Based on the numbers of cell edges, cell area and geometries of fitted ellipses, we derived and verified the new relations of Lewis’s law and Aboav-Weaire’s law. Lewis’s law for a n-edged cell: \[cell\ area=0.5nab\sin\left(\frac{2\pi}{n}\right)\left(1-\frac{3}{n^2}\right)\] Aboav-Weaire’s law: \[average\ side\ number\ of\ neighboring\ cells=6+\frac{6-n}{n}\times \left(\frac{a}{b}+\frac{3}{n^2}\right)\] where \(a\) and \(b\) are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.

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Ellipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law

10.7287/peerj.preprints.27421v1 ◽

2018 ◽

Author(s):

Kai Xu

Keyword(s):

Major Axis ◽

Theoretical Explanation ◽

Minor Effect ◽

Semi Major Axis ◽

Polygonal Cell ◽

Cell Topology ◽

Form Deviation ◽

A Minor ◽

Inscribed Polygons ◽

Ellipse Packing

Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear. Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis. Results: The average form deviation of vertexes from the fitted ellipse was 0±3.1 % (8,291 vertices in 1375 cells were examined). Area of the polygonal cell was 0.9±0.1 times of area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells can be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Then, an improved relation of Lewis’s law for a n-edged cell was derived \[cell\ area=0.5nab\sin(\frac{2\pi}{n})(1-\frac{3}{n^2})\] where, a and b are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. This study also improved the relation of Aboav-Weaire’s law \[number\ of\ neighboring\ cells=6+\frac{6-n}{n}(\frac{a}{b}+\frac{3}{n^2})\] Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.

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Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis’s law and Aboav-Weaire’s law

PeerJ ◽

10.7717/peerj.6933 ◽

2019 ◽

Vol 7 ◽

pp. e6933 ◽

Cited By ~ 1

Author(s):

Kai Xu

Keyword(s):

Growth Pattern ◽

Short Range ◽

Short Range Order ◽

Theoretical Explanation ◽

Cell Area ◽

Two Dimensional ◽

R Software ◽

Cell Topology ◽

Dimensional Cell ◽

Ellipse Packing

Background Lewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear. Methods We used R software with the Conicfit package to fit ellipses based on the geometric parameters of polygonal cells of ten different kinds of natural and artificial 2D structures. Results Our results indicated that the cells could be classified as an ellipse’s inscribed polygon (EIP) and that they tended to form the ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law. Conclusions Ellipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.

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Ellipse packing in two-dimensional celltessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law

10.7287/peerj.preprints.27421 ◽

2019 ◽

Author(s):

Kai Xu

Keyword(s):

Growth Pattern ◽

Short Range ◽

Short Range Order ◽

R Package ◽

Theoretical Explanation ◽

Geometric Parameters ◽

Cell Area ◽

Two Dimensional ◽

Cell Topology ◽

Ellipse Packing

Background: Lewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear. Methods: We used R package Conicfit software to fit ellipses based on the geometric parameters of polygonal cells with ten different kinds of natural and artificial 2D structures. Results: Our results indicated that the cells could be classified as ellipse’s inscribed polygon (EIP) and that they tended to form ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law . Conclusions: Ellipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.

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Ellipse packing in two-dimensional celltessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law

10.7287/peerj.preprints.27421v3 ◽

2019 ◽

Author(s):

Kai Xu

Keyword(s):

Growth Pattern ◽

Short Range ◽

Short Range Order ◽

R Package ◽

Theoretical Explanation ◽

Geometric Parameters ◽

Cell Area ◽

Two Dimensional ◽

Cell Topology ◽

Ellipse Packing

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The unexpected narrowness of eccentric debris rings: a sign of eccentricity during the protoplanetary disc phase

Royal Society Open Science ◽

10.1098/rsos.200063 ◽

2020 ◽

Vol 7 (6) ◽

pp. 200063

Author(s):

Grant M. Kennedy

Keyword(s):

Major Axis ◽

Semi Major Axis ◽

Ring Systems ◽

Secular Perturbations ◽

Standard Scenario ◽

Narrow Belt ◽

A Minor ◽

Protoplanetary Discs

This paper shows that the eccentric debris rings seen around the stars Fomalhaut and HD 202628 are narrower than expected in the standard eccentric planet perturbation scenario (sometimes referred to as ‘pericentre glow’). The standard scenario posits an initially circular and narrow belt of planetesimals at semi-major axis a , whose eccentricity is increased to e f after the gas disc has dispersed by secular perturbations from an eccentric planet, resulting in a belt of width 2 ae f . In a minor modification of this scenario, narrower belts can arise if the planetesimals are initially eccentric, which could result from earlier planet perturbations during the gas-rich protoplanetary disc phase. However, a primordial eccentricity could alternatively be caused by instabilities that increase the disc eccentricity, without the need for any planets. Whether these scenarios produce detectable eccentric rings within protoplanetary discs is unclear, but they nevertheless predict that narrow eccentric planetesimal rings should exist before the gas in protoplanetary discs is dispersed. PDS 70 is noted as a system hosting an asymmetric protoplanetary disc that may be a progenitor of eccentric debris ring systems.

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Detection and Characterization of earth-like Planets

International Astronomical Union Colloquium ◽

10.1017/s0252921100014810 ◽

1997 ◽

Vol 161 ◽

pp. 299-311 ◽

Cited By ~ 22

Author(s):

Jean Marie Mariotti ◽

Alain Léger ◽

Bertrand Mennesson ◽

Marc Ollivier

Keyword(s):

Direct Detection ◽

Contrast Ratio ◽

Angular Size ◽

Physical Nature ◽

Major Axis ◽

Infrared Emission ◽

Space Technology ◽

Semi Major Axis ◽

Earth Like Planets ◽

Visible Wavelengths

AbstractIndirect methods of detection of exo-planets (by radial velocity, astrometry, occultations,...) have revealed recently the first cases of exo-planets, and will in the near future expand our knowledge of these systems. They will provide statistical informations on the dynamical parameters: semi-major axis, eccentricities, inclinations,... But the physical nature of these planets will remain mostly unknown. Only for the larger ones (exo-Jupiters), an estimate of the mass will be accessible. To characterize in more details Earth-like exo-planets, direct detection (i.e., direct observation of photons from the planet) is required. This is a much more challenging observational program. The exo-planets are extremely faint with respect to their star: the contrast ratio is about 10−10at visible wavelengths. Also the angular size of the apparent orbit is small, typically 0.1 second of arc. While the first point calls for observations in the infrared (where the contrast goes up to 10−7) and with a coronograph, the latter implies using an interferometer. Several space projects combining these techniques have been recently proposed. They aim at surveying a few hundreds of nearby single solar-like stars in search for Earth-like planets, and at performing a low resolution spectroscopic analysis of their infrared emission in order to reveal the presence in the atmosphere of the planet of CO H2O and O3. The latter is a good tracer of the presence of oxygen which could be, like on our Earth, released by biological activity. Although extremely ambitious, these projects could be realized using space technology either already available or in development for others missions. They could be built and launched during the first decades on the next century.

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Improvement of monitoring of tertiary filtration with particle counting

Water Science & Technology Water Supply ◽

10.2166/ws.2006.001 ◽

2006 ◽

Vol 6 (1) ◽

pp. 1-9

Author(s):

V. Miska ◽

J.H.J.M. van der Graaf ◽

J. de Koning

Keyword(s):

Particle Surface ◽

Minor Effect ◽

Particle Size Distributions ◽

Particle Volume ◽

Mass Distributions ◽

Particle Counting ◽

Total Particle ◽

Particle Counts ◽

A Minor ◽

Sample Dilution

Nowadays filtration processes are still monitored with conventional analyses like turbidity measurements and, in case of flocculation–filtration, with phosphorus analyses. Turbidity measurements have the disadvantage that breakthrough of small flocs cannot be displayed, because of the blindness regarding changes in the mass distributions. Additional particle volume distributions calculated from particle size distributions (PSDs) would provide a better assessment of filtration performance. Lab-scale experiments have been executed on a flocculation–filtration column fed with effluent from WWTP Beverwijk in The Netherlands. Besides particle counting at various sampling points, the effect of sample dilution on the accuracy of PSD measurements has been reflected. It was found that the dilution has a minor effect on PSD of low turbidity samples such as process filtrate. The correlation between total particle counts, total particle volume (TPV) and total particle surface is not high but is at least better for diluted measurements of particles in the range 2–10 μm. Furthermore, possible relations between floc-bound phosphorus and TPV removal had been investigated. A good correlation coefficient is found for TPV removal versus floc-bound phosphorus removal for the experiments with polyaluminiumchloride and the experiments with single denitrifying and blank filtration.

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Linkage mapping identifies a non-synonymous mutation in FLOWERING LOCUS T (FT-B1) increasing spikelet number per spike

Scientific Reports ◽

10.1038/s41598-020-80473-0 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Jonathan Brassac ◽

Quddoos H. Muqaddasi ◽

Jörg Plieske ◽

Martin W. Ganal ◽

Marion S. Röder

Keyword(s):

Flowering Time ◽

Aspartic Acid ◽

Synonymous Substitution ◽

Minor Effect ◽

Phenotypic Variance ◽

Spikelet Number ◽

Wheat Varieties ◽

Trait Locus ◽

A Minor ◽

Spikelet Number Per Spike

AbstractTotal spikelet number per spike (TSN) is a major component of spike architecture in wheat (Triticumaestivum L.). A major and consistent quantitative trait locus (QTL) was discovered for TSN in a doubled haploid spring wheat population grown in the field over 4 years. The QTL on chromosome 7B explained up to 20.5% of phenotypic variance. In its physical interval (7B: 6.37–21.67 Mb), the gene FLOWERINGLOCUST (FT-B1) emerged as candidate for the observed effect. In one of the parental lines, FT-B1 carried a non-synonymous substitution on position 19 of the coding sequence. This mutation modifying an aspartic acid (D) into a histidine (H) occurred in a highly conserved position. The mutation was observed with a frequency of ca. 68% in a set of 135 hexaploid wheat varieties and landraces, while it was not found in other plant species. FT-B1 only showed a minor effect on heading and flowering time (FT) which were dominated by a major QTL on chromosome 5A caused by segregation of the vernalization gene VRN-A1. Individuals carrying the FT-B1 allele with amino acid histidine had, on average, a higher number of spikelets (15.1) than individuals with the aspartic acid allele (14.3) independent of their VRN-A1 allele. We show that the effect of TSN is not mainly related to flowering time; however, the duration of pre-anthesis phases may play a major role.

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Some Special Types of Orbits around Jupiter

Aerospace ◽

10.3390/aerospace8070183 ◽

2021 ◽

Vol 8 (7) ◽

pp. 183

Author(s):

Yongjie Liu ◽

Yu Jiang ◽

Hengnian Li ◽

Hui Zhang

Keyword(s):

Gravity Model ◽

Small Perturbation ◽

Equilibrium Points ◽

Major Axis ◽

Semi Major Axis ◽

Ground Track ◽

The Earth ◽

Degenerate Equilibrium ◽

The Mean ◽

Element Theory

This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

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