## 1 Introduction
Lett._ 1993, 70, 3915-3918. Copyright (1993) by the American Physical Society). thermal conductivity measured for QCs over the wide temperature interval 1-400 K [3; 6; 10], which could be explained in terms of an enhanced Umklapp process stemming from the dense filling of Bragg reflections in reciprocal space (Figure 1a). Notwithstanding this, as far as I know no definitive theoretical explanation has been given to the physical origin of QCs' specific properties on the sole basis of the very nature of the new kind of underlying order they possess. On the contrary, a growing number of experimental results and numerical simulations alike suggest that a main factor in the origin of the remarkable properties of both d- and i-QCs is probably related to factors involving nearest and next-nearest atomic neighbors, thereby highlighting the important role of short-range effects in the emergence of unusual physical properties in QCs [17; 18; 19; 20]. To gain some insight into this important issue it is convenient to note that, in most quasicrystalline forming alloy systems, the true QC phase is accompanied by a number of compositionally-related periodic crystals, having huge unit cell sizes (containing up to 150-200 atoms), which are called _approximant_ phases, because these crystals not only have very similar compositions, but also atomic structures closely resembling that of the true QC compound, from which they can nevertheless be distinguished. Approximant crystals are important for understanding the structure of the related QCs since their atomic arrangements can be determined with high accuracy, which is very helpful in obtaining a first-approach QC structure model. Within the hyperspace formalism both periodic and QP structures can be derived by projection from a parent hyperlattice onto the 3D physical space [3; 4], the main difference being that for QCs the projection involves irrational numbers, whereas in the case of approximant crystals the projection is given in terms of rational _approximant numbers_ to those irrational quantities. Accordingly, approximant crystals to i-QCs can be classified by a rational order number \(\mathrm{F_{n}/F_{n-1}}\), which is a ratio of two consecutive terms in the Fibonacci sequence given by the recurrence formula \(\mathrm{F_{n+1}=F_{n}+F_{n-1}}\), with \(\mathrm{F_{0}=F_{1}=1}\). Thus, from a numerical study on covalent bonding and semiconducting bandgap formation in quasicrystalline approximants bearing Al and transition metal (TM) atoms, it was concluded that it is not the long-range QPO but an unusual Al-TM bonding that is responsible for a substantial part of the peculiarities observed in the transport properties of this QC alloy system [21]. A similar conclusion was reached from ab-initio studies of i-AlPdMn QCs suggesting that, since the long-range QPO modifies only to a limited extent the features in the electronic DOS resulting from the short-range order, then the local chemistry could be a more dominant factor in determining the electronic properties of these QCs [22]. Alternatively, the non-metallic behavior of QCs could also be understood as being due to the QP arrangement of atoms throughout the space, which naturally leads to the formation of a deep pseudogap near the Fermi energy, along with the existence of a significant number of electronic states exhibiting very low diffusion coefficients. To this end, one must take into account the existence of a richer behavior for electronic states in QCs: On the one hand, we have extended electronic wave functions (Figure 3a) able to open a pseudogap close to the Fermi level via diffraction and interference processes with quasiperiodically-stacked arrangements of atomic planes; on the other hand, we have localized electronic states as well (Figure 3b), stemming from resonant effects involving nested atomic clusters exhibiting self-similar geometries at different scales.