Space of Nanoworld

Abstract

In this chapter, a geometrical model to accurately describe the distribution of light points in diffraction patterns of quasicrystals is proposed. It is shown that the proposed system of parallel lines has axes of the fifth order, periodically repeating the fundamental domain of the quasicrystals. A 4D- polytope, called the golden hyperrombohedron is introduced. It consists from eight rhombohedrons densely filling the 4D space (like the regular 8-Cell). Faces of the hyperrombohedron are connected by the golden section; they can be scaled as needed. On this universal lattice of the golden hyperrombohedron, famous crystallographic lattices: Bravais, Delaunay, Voronoi, etc. can be embedded. In supporting the idea of n-dimensional domains entangled within the three-dimensional Euclidean space, in minerals or synthetic chemicals, two series of small double-shell clusters are designed by operations on maps and their topological properties discussed.