Polygon edges

A two-dimensional network of cells consists of polygons, edges (sides), and comers. The number of each is governed by the simple relation... 

The sequence of 6x used for taking this limit is conveniently the sequence of polygon edges at successive refinements of the original polygon. 

Figure 31 d shows how the presence of a -n disclination eliminates very acute conical shapes of nested layers. This arrangement of layers creates a rhombic and conical domain in the midpart of most of the polygon edges in Fig. 6 a. Careful examination of the layers under natural light at a level close to the coverslip generally shows a structure like the one shown in Fig. 38 f. The disclination forms a helical half-loop [20] due to the cholesteric twist, a situation very similar to that described in other helical cholesteric patterns (see Sec. 7.3.4). 

In the polygonized nanotubes observed by Liu and Cowley]12,13], the edges of the polygon must have more sp character than the flat faees in between. These are defeet lines in the sp network. Nanotubes mechanically deformed appear to be rippled, indicat-... 

Vielheit, /. multiplicity multitude, viel-kantig, a. many-edged, polygonal, -ker-nig, a. polycyclic multinuclear, polynuclear. 

Figure 27. Seven possible cases of the polygonal surface representation in a single pyramid. The Euler characteristic is calculated as a sum of the number of faces and the number of vertices minus the number of edges of the polygons. The black and white circles represent points with higher and lower values relative to the threshold one. The gray area is the schematic representation of the surface inside a pyramid [225].

The curvature k of the interface in two dimensions is calculated in a similar way. Consider a polygon consisting of vertices v,- connected by edges Let us denote by /, the length of the edge between (7 — l)th and th vertex. The curvature k at the /th vertex can be then approximated as follows ... 

Fig. 1. Schematic cross-sectional views of SNF pellet showing an edge of an SNF with gap region, and a diagram of the grain structure showing e-particles, gas bubbles, and the polygonized rim region.

As a final artistic piece, consider Figure H.4 by Professor Carlo H. Sequin from the University of California, Berkeley. His representation is a projection of a 4-D 120-cell regular polytope (a 4-D analog of a polygon). This structure consists of twelve copies of the regular dodecahedron — one of the five Platonic solids that exist in 3-D space. This 4-D polytope also has 720 faces, 1200 edges, and 600 vertices, which are shared by two, three, and four adjacent dodecahedra, respectively. 

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