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 ).

n-dimensional enrichment for Further Mathematicians

2004 ◽  
Vol 89 (516) ◽  
pp. 409-416 ◽  
Author(s):  
Martin Griffiths
Keyword(s):  
Higher Dimensions ◽  
Three Dimensions ◽  
Regular Tetrahedron ◽  
Regular Polygons ◽  
Regular Polytopes ◽  
Platonic Solids ◽  
Mathematical Objects ◽  
Regular Octahedron ◽  
Regular Polyhedra

There are infinitely many regular polygons, but we find, on extending the idea of polygons to three dimensions, that there are only five regular polyhedra, the Platonic solids. What happens then if we try to extend this idea beyond three dimensions? It turns out that, of the five Platonic solids, just the regular tetrahedron, cube and regular octahedron have analogues in all higher dimensions, the so-called regular polytopes. Brief descriptions of these mathematical objects are to be found in [1], for example.

Soft template assisted self-assembly: a general strategy toward two-dimensional molecular crystals for high-performance organic field-effect transistors

2022 ◽  
Author(s):  
Xinzi Tian ◽  
Jiarong Yao ◽  
Siyu Guo ◽  
Zhaofeng Wang ◽  
Yanling Xiao ◽  
...  
Keyword(s):  
Organic Semiconductors ◽  
Self Assembly ◽  
High Performance ◽  
Field Effect Transistors ◽  
Molecular Crystals ◽  
Soft Template ◽  
General Strategy ◽  
Two Dimensional ◽  
Organic Field Effect Transistors ◽  
Intrinsic Properties

Two-dimensional molecular crystals (2DMCs) are highly desirable to probe the intrinsic properties in organic semiconductors and are promising candidates for constructing high-performance optoelectronic devices. Liquids such as water are favorable...

Extremum Properties of the Regular Polyhedra

1950 ◽  
Vol 2 ◽  
pp. 22-31 ◽  
Author(s):  
Lâszlό Fejes Tόth
Keyword(s):  
Regular Polygons ◽  
Surface Areas ◽  
Extremum Problems ◽  
Regular Polyhedra ◽  
Historical Remarks

1. Historical remarks. In this paper we extend some well-known extremum properties of the regular polygons to the regular polyhedra. We start by mentioning some known results in this direction.First, let us briefly consider the problem which has received the greatest attention among all the extremum problems for polyhedra. It is the determination of the polyhedron of greatest volume F of a class of polyhedra of equal surface areas F, i.e., the isepiphan problem.

scholarly journals Skull asymmetries in wild boar (Sus scrofa LINNAEUS, 1758)

2019 ◽  
Vol 56 (1) ◽  
pp. e150704
Author(s):  
Pere M. Parés-Casanova
Keyword(s):  
Environmental Factors ◽  
Wild Boar ◽  
Sus Scrofa ◽  
Age Groups ◽  
The Other ◽  
Directional Asymmetry ◽  
Two Dimensional ◽  
Entire Sample ◽  
Perfect Symmetry ◽  
Left And Right

Organisms can develop different kinds of asymmetry when deviations from expected perfect symmetry occur. Among others are fluctuating asymmetry (FA) and directional asymmetry (DA). FA represents small random differences between corresponding parts on the left and right sides of an individual in bilaterally paired structures. It is thought that FA reflects an organism’s ability to cope with genetic and environmental stress during growth. DA occurs whenever one side on the plane of symmetry develops more than the other side, and has a genetic component. In this research, we examined the expression of morphological symmetry in 38 skulls of different age groups of wild boar (Sus scrofa), on their ventral aspect, using two-dimensional coordinates of 27 landmarks. Analyses showed the presence of significant FA and DA in the entire sample, detecting also distinctive differences between age groups. The obtained results show that the shape differences in different age groups could reasonably be a consequence of a response to environmental factors for FA and a masticatory lateralization for DA.

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 $\widehat {\mathbb {C}}\setminus Q_n$ .

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

Historically Speaking— A Geometry Capsule Concerning the Five Platonic Solids

1969 ◽  
Vol 62 (1) ◽  
pp. 42-44
Author(s):  
Howard Eves
Keyword(s):  
Regular Polygons ◽  
Platonic Solids ◽  
Regular Polyhedra ◽  
Number Of Faces

A polyhedron is said to be “regular” if its faces are congruent regular polygons and its polyhedral angles are all congruent. While there are regular polygons of all orders, it is surprising that there are only five different regular polyhedra. These regular polyhedra have been named according to the number of faces each possesses. Thus there is the tetrahedron with four triangular faces, the hexahedron (cube) with six square faces, the octahedron with eight triangular faces, the dodecahedron with twelve pentagonal faces, and the icosahedron with twenty triangular faces. See the accompanying figure.

ELECTRONIC STRUCTURE AND FERMI SURFACE OF THE QUATERNARY INTERMETALLIC BOROCARBIDE SUPERCONDUCTOR YNi2B2C FROM 2D-ACAR

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5973-5982 ◽  
Author(s):  
A. S. HAMID
Keyword(s):  
Electronic Structure ◽  
Fermi Surface ◽  
Fourier Transformation ◽  
High Temperature Superconductors ◽  
Complex Surface ◽  
Momentum Density ◽  
Annihilation Radiation ◽  
Two Dimensional ◽  
Superconducting Properties ◽  
Topological Information

We measured the angular momentum density distribution of YNi 2 B 2 C to acquire information about its electronic structure. The measurements were performed using the full-scale utility of the two-dimensional angular correlation of annihilation radiation (2D-ACAR). The measured spectra clarified that Ni (3d) like state, predominantly, affected the Fermi surface of YNi 2 B 2 C . Further, s- and p-like-states enhanced its superconducting properties. The Fermi surface of YNi 2 B 2 C . was reconstructed using Fourier transformation followed by the LCW (Loucks, Crisp and West) folding procedure. It showed a large and complex surface similar to that of the high temperature superconductors HTS, with anisotropic properties. It also disclosed the effect of d-like state. Nevertheless, the current Fermi surface could deliver the needed topological information to isolate its features. The general layouts of this Fermi surface are; two large electron surfaces running along Γ–Z direction; as well as an additional large electron surface centered on X point; beside one hole surface centered on 100 point. This Fermi surface was interpreted in view of the earlier results.

ChemInform Abstract: General Strategy for Zero-Valent Intercalation into Two-Dimensional Layered Nanomaterials.

ChemInform ◽  
2014 ◽  
Vol 45 (25) ◽  
pp. no-no
Author(s):  
Janina P. Motter ◽  
Kristie J. Koski ◽  
Yi Cui
Keyword(s):  
General Strategy ◽  
Two Dimensional
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