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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
The dual polyhedron can be constructed by connecting the midpoints of every face of the original polyhedron, which thereby becomes the vertices of the dual polyhedron. Starting with any given polyhedron, the dual of its dual is the original polyhedron. For example, if the midpoints of each of the six square faces of a cube having eight vertices are joined by a line, the resulting polyhedron is an octahedron having six vertices and eight faces. Thus, the octahedron is the dual solid of the cube and vice versa. If we do not directly connect the central points of each polygonal face but rather the points lying vertically above or below them, we arrive at the dual of the polyhedron that now lies completely outside or inside the original one. Going on that way, we can construct a shell structure polyhedron@dual@polyhedron@dual and so on. This is a possible symmetry preserving building principle to contrive onionlike inorganic fullerenes. In Section 15.4, this is outlined in greater detail. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetry groups, the duals belong to the same symmetry group. Thus, e.g. the regular polyhedra – the (convex) Platonic solids – form dual pairs between themselves, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal. Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron (cf. Alexandrov 2005). A summary of the most common classes of polyhedra that also appear in molecular arrangements is given in the following Table 15.1 (see also Appendix).
Primal and dual algorithms for optimization over the efficient set
Published in Optimization, 2018
Zhengliang Liu, Matthias Ehrgott
In Section 5, we reached the conclusion that an optimal solution to Problem (8) can be found at an incomplete vertex of . An analogous idea applies to the facets of the dual polyhedron. In the rest of this section, we develop this idea through the association between the upper image of the primal MOLP and the lower image of the dual MOLP and exploit it to propose the dual algorithm.