Icosahedral addition patterns

The optimal styx pattern of three ctbb, ten Tbbb, and twelve OBH(t) bonds requires a total of 2(3 + 10 + 12) = 50 electrons, whereas a neutral (BH)i2 framework provides only 12(3 + 1) = 48 valence electrons. Hence, two additional electrons must be added to form the B12H122- anion that achieves the idealized icosahedral bonding pattern (3.259). Figure 3.111 depicts the computed structure of Bi2Hi22, which indeed optimizes to perfect icosahedral symmetry with each B—B = 1.787 A (experimental 1.77 A) and B—H = 1.202 A. 

L. Pauling, Additional evidence from x-ray powder diffraction patterns that icosahedral quasicrystals of intermetallic compounds are twinned cubic crystals. Proc. Natl. Acad. Sci. (USA) 85, 4587-4590 (1988). 

Whether quasicrystalline structures are limited to alloys remains an open question. It is possible that their occurrence is much more widespread than had been previously thought. Indeed there is evidence for quasicrystallinity in both thermotropic and lyotropic liquid crystals. Diffraction patterns of decagonal symmetry have been recorded in lyotropic liquid crystals [K. Fontell, private communication], (Fig. 2.19), and there is theoretical evidence for the existence of a quasicrystalline structure within the blue phase of cholesterol (Chapters 4, 5). (The decagonal structure has quasisymmetry perpendicular to the tenfold axes, and translation symmetry along them.) Viruses crystallise in icosahedral clusters and the list continues to grow. In addition to five-fold symmetry, it has been shown that eight and ten- fold quasisymmetry is possible. ... 

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