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Onion-Like Inorganic Fullerenes from a Polyhedral Perspective
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2020
Ch. Chang, A. B. C. Patzer, D. Sülzle, H. Bauer
where × is called the Euler characteristic and g the genus (i.e. the number of holes in or tunnels through the polyhedral framework). The Euler characteristic is a topological invariant that describes the shape of a structure regardless of how it is bent. Positive/negative × values indicate positive/negative curvature of the polyhedral structure. For every convex polyhedron, i.e. a polyhedron having no holes, tunnels, kinks etc. the genus g is equal to 0 and, therefore, χ is always equal to 2 and the Euler relation1 becomes: υ−e+f=2.
Graphs and Surfaces
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
We know from $$$xref rid="chapter12" ref-type="book-part">chapter 12%%%/xref$$$ that a connected planar graph with n vertices, ε edges, and f faces satisfies Euler’s formula n — ε + f = 2. Furthermore, the skeleton of a polyhedron is a planar graph, so that any polyhedral division of the sphere also satisfies this formula. We say that the Euler characteristic of the sphere is 2.
Topological perception on attention to product shape
Published in International Journal of Design Creativity and Innovation, 2020
Topology is a major branch of mathematics concerned with spatial properties that are preserved under one-to-one, continuous transformations, such as stretching and bending, but not breaking or fusing. Important topological properties include connectedness and compactness. Hence, solid figures such a cube and a tetrahedron are topologically equivalent, because one can be transformed into the other through continuous transformations. An attribute of an object is called topologically invariant if it does not change under a continuous transformation. If two objects have different topological invariants, they must be different homeomorphisms. One of the most common topological invariants is the Euler characteristic, which is commonly denoted by χ and conventionally defined for the surfaces of a polyhedron as χ= V – E + F, where V, E, and F are the numbers of vertices (corners), edges, and faces in the polyhedron, respectively. For example, the surface of a convex polyhedron has the Euler characteristic χ = V – E + F = 2. On the other hand, several human-made goods are homeomorphic to either spheres or rings and can be molded forming tiles. The topological transformations can be pictured as distorting a rubber sheet without creating holes or fusing edges. The Euler characteristics of a sphere and a ring (i.e. a torus) are 2 and 0, respectively.
Design in the service of topology and the classification theorem
Published in International Journal of Mathematical Education in Science and Technology, 2023
Note that for any given map on the sphere, the Euler characteristic number of a sphere is 2: it does not have an end, it is orientable, and all the surfaces that maintain these properties (as in Theorem 3.5) are homeomorphic.
The Dirac equation as a model of topological insulators
Published in Philosophical Magazine, 2020
Xiao Yuan, M. Bowen, P. S. Riseborough
The Gauss–Bonnet Theorem links differential geometry to the topology. It states that for a two-dimensional manifold with a boundary the Euler characteristic is given bywhere κ is the curvature and is the geodesic curvature. The Euler characteristic for a surface is a topological invariant given bywhere g, the genus, is equal to the number of holes. The Chern Theorem [26,27] is a generalisation of the Gauss–Bonnet Theorem to -dimensional manifolds. The Chern number for a band indexed by τ is related to the Berry phase. The Berry phase is an integral of the Berry curvature over an open surface, ie one that has a boundary. The Chern number (of the first kind) is defined as an integral over a closed, orientable, two-dimensional manifold and leads to integer topological invariants. For a crystalline lattice for which Bloch's Theorem applies, an integral over a two-dimensional Brillouin zone is equivalent to an integral over the closed surface of a torus (see Figure 8). For a solid which is time-reversal invariant the Chern number is zero. A non-zero Chern number indicates that the wave function does not have a smoothly varying and uniquely defined phase at each point of the closed surface. This may happen whenever the wave function goes to zero at a point, since the Berry phase of a closed contour encircling the point may change by multiples of . Hence, the Chern number describes the net number of vortices passing through the surface. For a manifold with boundaries, the integral of the Berry curvature is equal (modulo ) to the sum of the Berry phases around the boundaries. However, each Berry phase contribution from a boundary has an equivalent contribution from the Kramer's conjugate boundary. Hence, the index, through the Atiyah–Singer index Theorem [28], can be defined in terms of an integration over half the area of the Brillouin zone and the boundaries around the singularities. The volume of integration is chosen such that and its Kramer's conjugate point are never included in the area. The index is given byHence, has values of .