Icosahedral sphere

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

The general geometrical problem of the packing of spheres has not been solved. An example of closest packing of atoms with some variation in effective radius is the icosahedral packing found (13) in the intermetallic compound Mg3B(Al,Zn) (Fig. 1). The successive layers in this structure contain 1, 12, 32, and 117 spheres. These numbers are reproduced (to within 1) by the empirical equation (12)... 

I have assumed that this equation applies to structures with two or more spheres in the central layer (as well as with one. as in icosahedral packing), and have applied it in the calculation of the ranges of values of the neutron number N in which successive subsubshells are occupied (12). (In this calculation the difference in radius of the different kinds of spherons is taken into consideration.) The assignment of quantum numbers is made with use of the following assumptions (14) ... 

Bergman- and Tsai-type clusters have the same geometric types for the second, third, and fifth shells, with the innermost and the penultimate shells being different. Particularly, the third (icosahedral) shell and the outmost triacontahedral shell define comparably sized spheres for both, but the other three shells for Tsai types are about 1.0 A smaller in diameter than for Bergman types. 

Schematic illustration of the icosahedral rhinovirus 14. (a) Shown is the icosahedron comprised of 60 copies each of VP1 (light gray), VP2 (black), and VP3 (gray). The shaded circles around each five-fold axis indicate the canyon positions. Also indicated is the approximate position of the VP1 hydrophobic pocket that lies underneath the surface of the virion, (b) An icosahedral pentamer is expanded with one viral protomer shown as a protein ribbon diagram, (c) This pentamer is seen in a cutaway view. Here VP1 is white, VP2 and VP4 black, and VP3 gray. A capsid-binding compound is depicted as black spheres inside the VP1 ribbon diagram. The cross hatched regions on the (c) schematic (right) indicate areas that disorder when HRV14 crystals are exposed to acid.

In this chapter, which is an adaptation of [DeDeOO], are considered icosahedral fulleroids (or I-fulleroids, or, more precisely, 1(5, byfulleroids, i.e. ( 5, b], 3)-spheres of symmetry I or / ). For some values of b, the smallest such fulleroids are indicated and their unicity is proved. Also, several infinite series of them are presented. 

The second main subject - fc-valent two-faced maps - is treated in Chapter 3 and Chapters 9-19. Chapter 3 deals with our main example, fullerenes, while Chapter 9 classifies strictly face-regular maps on sphere or torus. In Chapters 10-18, we consider a weaker notion of face-regularity. Chapter 19 treats 3-valent two-faced maps with icosahedral symmetry. 

At the present time, there is no accepted chelating agent which can be used against common influenza viruses in humans. A virus has a core of either DNA or RNA and a protective coat of many identical protein units. All viruses are either rods or spheres, that is the protein coats are cylindrical shells having helical symmetry or spherical shells having icosahedral symmetry. Viruses reproduce inside living cells, where each viral nucleic acid directs the synthesis of about 1000 fresh viruses. These are then released and the host cell may die. 

An initial approach to fullerene enumeration was based on point-group symmetry (Fowler 1986 Fowler et al. 1988) and involved an extension of Coxeter s (1971) work on icosahedral tessellations of the sphere and of methods for the classification of virus structures (Caspar Klug 1962). This approach led to magic numbers in fullerene electronic structure (Fowler Steer 1987 Fowler 1990) and will be described briefly here. 

The Pb2 atoms have distorted icosahedral holmium coordination. These icosa-hedra show an orthorhombic body-centered packing. All three crystallographically independent holmium atoms are in the coordination sphere of the Pb2 atoms. 

The length of an edge of a regular icosahedron is some 5% greater than the distance from the center to vertex. Thus, the sphere of the outer shell of 12 makes contact only with the central sphere. Conversely, if each sphere of an icosahedral group of 12, all touching the central sphere, is in contact with its 5 neighbors, then the central sphere must have a radius of some 10% smaller than the radius of the outer spheres. The relative size considerations are important in the... 

Figure 9-30. Icosahedral packing of spheres showing the third shell [53], This is popularly called the Mackay icosahedron. ...

Applying ab initio quantum-chemical methods and density functional theory in the local density approximation, different (BH) spherical clusters for n — 12,20,32,42 and 92 have been investigated. Most of the clusters show nearly icosahedral symmetry. The hydrogen atoms are bonded to the spherical surface as prickles. The relative stability of the spheres measured as the binding energy per molecule has been analyzed. All the clusters studied are very stable, and the spherical (BH)32 cluster Seems to be the most stable structure. The effect of the hydrogen atoms is to increase the stability of the bare boron clusters. 

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