scholarly journals Shape and Symmetry Determine Two-Dimensional Melting Transitions of Hard Regular Polygons

2017 ◽  
Vol 7 (2) ◽  
Author(s):  
Joshua A. Anderson ◽  
James Antonaglia ◽  
Jaime A. Millan ◽  
Michael Engel ◽  
Sharon C. Glotzer
Keyword(s):  
Two Dimensional ◽  
Regular Polygons ◽  
Melting Transitions

STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

2020 ◽  
pp. 1-12
Author(s):  
PENG-FEI ZHANG ◽  
XIN-HAN DONG
Keyword(s):  
Lebesgue Measure ◽  
Cauchy Transform ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Cauchy Transforms ◽  
Dimensional Lebesgue Measure

Abstract For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Stokes Eigenmodes on two-dimensional regular polygons

2021 ◽  
pp. 105069
Author(s):  
Pierre Lallemand ◽  
Lizhen Chen ◽  
Gérard Labrosse ◽  
Li–Shi Luo
Keyword(s):  
Two Dimensional ◽  
Regular Polygons ◽  
Stokes Eigenmodes

scholarly journals Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

PeerJ ◽  
2017 ◽  
Vol 5 ◽  
pp. e3314 ◽  
Author(s):  
Kai Xu ◽  
Yan Xu ◽  
Dehua Ji ◽  
Ting Chen ◽  
Changsheng Chen ◽  
...  
Keyword(s):  
Cell Proliferation ◽  
Cell Division ◽  
Light Microscope ◽  
Cell Walls ◽  
Single Layer ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Cell Divisions ◽  
Stages Of Growth ◽  
Perfect Model

Background Pyropia haitanensis thalli, which are made of a single layer of polygonal cells, are a perfect model for studying the morphogenesis of multi-celled organisms because their cell proliferation process is an excellent example of the manner in which cells control their geometry to create a two-dimensional plane. Methods Cellular geometries of thalli at different stages of growth revealed by light microscope analysis. Results This study showed the cell division transect the middle of the selected paired-sides to divide the cell into two equal portions, thus resulting in cell sides ≥4 and keeping the average number of cell sides at approximately six even as the thallus continued to grow, such that more than 90% of the cells in thalli longer than 0.08 cm had 5–7 sides. However, cell division could not fully explain the distributions of intracellular angles. Results showed that cell-division-associated fast reorientation of cell sides and cell divisions together caused 60% of the inner angles of cells from longer thalli to range from 100–140°. These results indicate that cells prefer to form regular polygons. Conclusions This study suggests that appropriate cell-packing geometries maintained by cell division and reorientation of cell walls can keep the cells bordering each other closely, without gaps.

scholarly journals Stereographic Visualization of 5-Dimensional Regular Polytopes

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 391
Author(s):  
Xingchang Wang ◽  
Tao Yu ◽  
Kwokwai Chung ◽  
Krzysztof Gdawiec ◽  
Peichang Ouyang
Keyword(s):  
Stereographic Projection ◽  
General Strategy ◽  
Two Dimensional ◽  
Regular Polygons ◽  
Regular Polytopes ◽  
Edge Information ◽  
Topological Information ◽  
Regular Polyhedra ◽  
Perfect Symmetry ◽  
Topological Data

Regular polytopes (RPs) are an extension of 2D (two-dimensional) regular polygons and 3D regular polyhedra in n-dimensional ( n ≥ 4 ) space. The high abstraction and perfect symmetry are their most prominent features. The traditional projections only show vertex and edge information. Although such projections can preserve the highest degree of symmetry of the RPs, they can not transmit their metric or topological information. Based on the generalized stereographic projection, this paper establishes visualization methods for 5D RPs, which can preserve symmetries and convey general metric and topological data. It is a general strategy that can be extended to visualize n-dimensional RPs ( n > 5 ).

Effects of impurity on two-dimensional melting transitions

1998 ◽  
Vol 57 (15) ◽  
pp. 9270-9273 ◽  
Author(s):  
Jian Ma ◽  
Eleanor D. Carter ◽  
Hillary B. Kleinberg
Keyword(s):  
Two Dimensional ◽  
Melting Transitions
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