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

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.

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

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.

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

PeerJ ◽  
2019 ◽  
Vol 7 ◽  
pp. e6933 ◽  
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.

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

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.

Antiferromagnetic Short Range Order in a Two-Dimensional Manganite Exhibiting Giant Magnetoresistance

1997 ◽  
Vol 78 (16) ◽  
pp. 3197-3200 ◽  
Author(s):  
T. G. Perring ◽  
G. Aeppli ◽  
Y. Moritomo ◽  
Y. Tokura
Keyword(s):  
Short Range ◽  
Giant Magnetoresistance ◽  
Short Range Order ◽  
Range Order ◽  
Two Dimensional

Effects of short-range order and interfacial interactions on the electronic structure of two-dimensional antimony-arsenic alloys

2020 ◽  
Vol 127 (2) ◽  
pp. 025305
Author(s):  
Qi An ◽  
Matthieu Fortin-Deschênes ◽  
Guanghua Yu ◽  
Oussama Moutanabbir ◽  
Hong Guo
Keyword(s):  
Electronic Structure ◽  
Short Range ◽  
Short Range Order ◽  
Range Order ◽  
Two Dimensional ◽  
Interfacial Interactions

Short-range ordering of Cu3Au above Tc in the topmost 80 A of a (001) face

1997 ◽  
Vol 12 (1) ◽  
pp. 75-82 ◽  
Author(s):  
M. Kimura ◽  
J. B. Cohen ◽  
S. Chandavarkar ◽  
K. Liang
Keyword(s):  
Short Range ◽  
Short Range Order ◽  
Surface Region ◽  
Antiphase Domain ◽  
Range Order ◽  
Two Dimensional ◽  
X Ray Diffraction ◽  
Diffuse Intensity ◽  
Near Surface ◽  
Atomic Displacements

The short-range order in the near surface region of the Cu3Au(001) face was investigated above the critical temperature by glancing-incidence x-ray diffraction, measuring the diffuse intensity throughout a two-dimensional region of reciprocal space. This intensity was analyzed quantitatively to obtain the two-dimensional Cowley–Warren short-range-order parameters and atomic displacements. Monte-Carlo simulation based on these values has revealed that the atomic configurations in the surface consist of ordered domains and clusters in a disordered matrix. There is a large number of {10} antiphase domain boundaries (APDB).

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