Euler characteristic surface topology

For diffuse and delocahzed interfaces one can still define a mathematical surface which in some way describes the film, for example by 0(r) = 0. A problem arises if one wants to compare the structure of microemulsion and of ordered phases within one formalism. The problem is caused by the topological fluctuations. As was shown, the Euler characteristic averaged over the surfaces, (x(0(r) = 0)), is different from the Euler characteristics of the average surface, x((0(r)) = 0), in the ordered phases. This difference is large in the lamellar phase, especially close to the transition to the microemulsion. x((0(r)) =0) is a natural quantity for the description of the structure of the ordered phases. For microemulsion, however, (0(r)) = 0 everywhere, and the only meaningful quantity is (x(0(r) = 0))-... 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

The main properties of a polyhedral surface are undoubtedly related to the topology of the underlying manifold that it decorates. This topology can be completely described by two properties the Euler characteristic and the orientability of the surface. The former can easily be calculated from the celebrated Euler theorem [9] which states that the number of vertices, edges, and faces, denoted as V, E, and F respectively, obey the following mle ... 

Here is a fixed integer, the Euler characteristic that marks the particular topology of the surface on which the polyhedron is embedded. However, in order to describe the topology completely, one also has to specify the orientability of the surface [3]. A surface is orientable if there is no walk on the surface that would take you from the outside to the inside. Such is the case of a sphere with handles. Otherwise, it is non-orientable. This is the case of a sphere with crosscaps. Based on this orientability, the infinite class of surfaces can be divided into two subclasses ... 

Suppose a two-dimensional vector field n is defined on a closed surface with Euler characteristic E. This field might contain point defects whose topological charges are defined as... 

This property holds because (F-E+V) is a topological characteristic, dependent only on the topology of the facetted surface. Since all polyhedra are topologically equivalent to the sphere (Fig 1.9), F-E+V) is conserved. The value of this integer is known as the Euler-Poincare characteristic x x) ... 

This result implies that the ring size and connectivity of a network determine the topology of the surface which contains that network. This allows for simple characterisation of cage, sheet and framework nets, distinguishable by the value of their Euler-Poincare characteristic (Table 1.2). 

Another topological characteristic, the genus, g(x), of a surface, is a measure of its connectedness. It is equal to the number of holes or handles in the surface and simply related to the Euler-Poincar characteristic by... 

The integral over the Gaussian curvature in Eq. [27] is a topological invariant.i "i85 For a closed orientable 2D surface (i.e., one without boundary), the Gauss—Bonnet theorem ties the value of this invariant to the genus g or the Euler—Poincare characteristic x of the surface ... 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

Besides the surface curvatures that define the metric relationships of objects, there are some fundamental aspects of shapes which are preserved if the structures are made of stretchable rubber sheet —topology. Suppose a surface is subdivided into a number of faces, edges, and vertices. A topological characteristic, called the Euler-Poincar characteristic, depends only on the topology of the facetted surface and is defined ... 

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