## 5 Quasicrystals as Cluster Aggregates  
These building units generally adopt well-defined polyhedral shapes and can be described as regular arrangements of atoms in nested shells adopting point group icosahedral symmetries (dodecahedron, icosahedron, triacontahedron, icosidodecahedron). By attending the detailed structure of these polyhedral i-QCs, they can be classified into three main classes, namely: (1) The AlMnSi class, whose prototype is given by the \(\alpha\)-Mn\({}_{12}\)(Al,Si)\({}_{57}\) phase; (2) the MgAlZn class, whose prototype is given by the Mg\({}_{32}\)(Al,Zn)\({}_{49}\) Bergman phase; and (3) the CdYb class, whose prototype is given by the Cd\({}_{6}\)Yb phase. In all cases the entire self-similar atomic structure can be obtained starting from a given polyhedral seed by systematically applying an inflation process via successive substitutions of atoms by clusters. The inflation operation consists of a proper rescaling by a scale factor \(\tau^{3}\), where \(\tau\) = (1 + \(\sqrt{5}\))/2 is the so-called golden mean. An example of the resulting structure is shown in Figure 8b, displaying an icosidodecahedral aggregate consisting of icosidodecahedral atomic clusters.  
As it deduced from X-ray and neutron diffraction data the structure of i-AlPdMn and i-AlCuFe QCs is based on the so-called pseudo-Mackay cluster, which contains 51 atoms and can be described in terms of three centrosymmetric atomic shells: A core of nine atoms, an intermediate icosahedron of 12 atoms, and an external icosidodecahedron of 30 atoms (Figure 8a) [46; 47; 48; 49]. The structures of stable i-QCs in the AlCuLi, ZnMgZr, ZnMgRE, ZnMg(Al,Ga) and MgAlPd systems were grouped in the MgAlZn class, which is based on the so-called Bergman clusters: The center is vacant, the first shell is an Al/Zn icosahedron, the second shell is a Mg dodecahedron, and the third shell is a larger Al/Zn icosahedron. Finally, stable i-QCs found in the binary Cd(Yb,Ca) and the ternary ZnScM, (Ag,Au)In(Yb,Ca), and CdMgRE alloy systems have been grouped in the CdYb Tsai-cluster third class, which constitutes the largest of the three classes of i-QCs. In the core of each Tsai-type cluster there is a tetrahedron created by four positionally-disordered Cd atoms, the first shell is a Cd dodecahedron, the second shell is a Yb icosahedron, and the third shell is a Cd icosidodecahedron [50; 51].  
A major question in the field is whether clusters are physically-significant, chemically-stable entities, or simply convenient geometrical constructions. A number of evidences supporting the existence of clusters as stable physicochemical entities have been reported in the literature, including secondary electron imaging [52], X-ray photoelectron diffraction [53], and scanning tunneling microscopy techniques [54]. The obtained results support the picture of i-QCs as cluster aggregates, which can be described in terms of a three-dimensional QP lattice properly decorated by atomic clusters having the same point symmetry as the whole QC [55]. This scenario naturally brings about some questions, which are the focus of intensive current research. For example, a given cluster may act as a chemically-stable structure when isolated, but it may lose its identity, due to strong interactions with close neighbors, when assembled to form a solid. So, what is the nature of the chemical bonding among the atomsbelonging to a given cluster, as well as among different clusters themselves? Then, along with the stability of clusters we should also consider those aspects related to their reactivity. In a similar way, one may wonder about the number and structure of the different atomic clusters compatible with the system's stoichiometry, or regarding the more appropriate packing rules involving atomic clusters at different hierarchical stages.