Euclidean planes in three-dimensional space

Résumé

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space \mathbb{R}^3. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. While a pair of real numbers \mathbb{R}^2 suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space \mathbb{R}^3. Derived concepts A plane segment (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment. A bivector is an oriented plane segment, analogous to directed line segments. A face is a plane segment bounding a solid object. A slab is a region bounded by two parallel planes. A parallelepiped is a region bounded by three pairs of parallel planes. Occurrence in nature Surface#In the physical sciences A plane serves a

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https://en.wikipedia.org/wiki/Euclidean_planes_in_three-dimensional_space

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Robustness against disorder and defects is a pivotal advantage of topological systems, manifested by the absence of electronic backscattering in the quantum-Hall and spin-Hall effects, and by unidirectional waveguiding in their classical analogues. Two-dimensional (2D) topological insulators, in particular, provide unprecedented opportunities in a variety of fields owing to their compact planar geometries, which are compatible with the fabrication technologies used in modern electronics and photonics. Among all 2D topological phases, Chern insulators are currently the most reliable designs owing to the genuine backscattering immunity of their non-reciprocal edge modes, brought via time-reversal symmetry breaking. Yet such resistance to fabrication tolerances is limited to fluctuations of the same order of magnitude as their bandgap, limiting their resilience to small perturbations only. Here we investigate the robustness problem in a system where edge transmission can survive disorder levels with strengths arbitrarily larger than the bandgap—an anomalous non-reciprocal topological network. We explore the general conditions needed to obtain such an unusual effect in systems made of unitary three-port non-reciprocal scatterers connected by phase links, and establish the superior robustness of anomalous edge transmission modes over Chern ones to phase-link disorder of arbitrarily large values. We confirm experimentally the exceptional resilience of the anomalous phase, and demonstrate its operation in various arbitrarily shaped disordered multi-port prototypes. Our results pave the way to efficient, arbitrary planar energy transport on 2D substrates for wave devices with full protection against large fabrication flaws or imperfections.

2021

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Representing shape and transparency in alpha planes

Laurent Piron

EPFL1999

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Lower bounds on the obstacle number of graphs

Padmini Mukkamala, János Pach, Doemoetoer Palvoelgyi

Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It is shown that there are graphs on n vertices with obstacle number at least Omega(n/logn).

2012

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