Hypothetical graphite structures with negative gaussian curvature

1993 ◽  
Vol 343 (1667) ◽  
pp. 113-127 ◽  
Keyword(s):  
Constant Curvature ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Hyperbolic Plane ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Three Dimensions ◽  
Ring Size ◽  
Two Dimensional ◽  
Planar Lattice

We consider the geometries of hypothetical structures, derived from a graphite net by the inclusion of rings of seven or eight bonds, which may be periodic in three dimensions. Just as the positive curvature of fullerene sheets is produced by the presence of pentagons, so negative curvature appears with a mean ring size of more than six. These structures are based on coverings of periodic minimal surfaces, and surfaces parallel to these, which are known as exactly defined mathematical objects. In the same way that the cylindrical and conical structures can be generated (geometrically) by curving flat sheets so that the perimeter of a ring can be identified with a vector in the two-dimensional planar lattice, so these structures can be related to tessellations of the hyperbolic plane. The geometry of transformations at constant curvature relates various surfaces. Some of the proposed structures, which are reviewed here, promise to have lower energies than those of the convex fullerenes

Negative Gaussian curvature induces significant suppression of thermal conduction in carbon crystals

Nanoscale ◽  
2017 ◽  
Vol 9 (37) ◽  
pp. 14208-14214 ◽  
Author(s):  
Zhongwei Zhang ◽  
Jie Chen ◽  
Baowen Li
Keyword(s):  
Gaussian Curvature ◽  
Thermal Conduction ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Significant Suppression ◽  
Carbon Allotropes ◽  
Zero Curvature ◽  
Negative Gaussian Curvature

From the mathematic category of surface Gaussian curvature, carbon allotropes can be classified into three types: zero curvature, positive curvature, and negative curvature.

scholarly journals Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces

2014 ◽  
Vol 14 (3) ◽  
pp. 393-409
Author(s):  
Régis Straubhaar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Laplace Operator ◽  
Numerical Optimization ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
The Laplace Operator

Abstract.Let (M,g) be a smooth and complete surface, $\Omega \subset M$ be a domain in M, and $\Delta _g$ be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence $0 < \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots \nearrow \infty $. A classical question is to ask what is the domain $\Omega ^*$ which minimizes $\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding $\lambda _m(\Omega _m^*)$. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for $\Omega _m^*$. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.

scholarly journals Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests

2015 ◽  
Vol 71 (6) ◽  
pp. 599-611 ◽  
Author(s):  
Myfanwy E. Evans ◽  
Stephen T. Hyde
Keyword(s):  
Minimal Surfaces ◽  
Hyperbolic Plane ◽  
Two Dimensional ◽  
Acta Cryst ◽  
Triply Periodic Minimal Surfaces ◽  
Triply Periodic ◽  
Current Classification ◽  
Metal Organic ◽  
Insight Into ◽  
Gaining Insight

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane ({\bb H}^2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in {\bb H}^2. This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal–organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given inPeriodic entanglementParts I and II [Evanset al.(2013).Acta Cryst.A69, 241–261; Evanset al.(2013).Acta Cryst.A69, 262–275].

Non-Euclidean Extensions

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Positive Curvature ◽  
Spherical Geometry ◽  
Two Dimensional ◽  
Actual Infinity ◽  
Constant Positive Curvature ◽  
Negative Constant ◽  
The Given ◽  
Line P

This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.

Sign in / Sign up

Export Citation Format

Share Document