Regular icosahedron

The 4 point group is that to which a regular icosahedron, illustrated in Figure 4.13(a), belongs. It contains 20 equilateral triangles arranged in a three-dimensional sttucture. This is the conformation of the anion, in which there is a boron atom, with a hydrogen atom... 

The configuration of the twelve ligands about the small MnlV atom is that of an approximately regular icosahedron. The thirteen-cornered and sixteen-cornered coordination polyhedra about Mnl, Mnll, and Mnlll are appropriate to axial ratios slightly greater than unity. 

Fig. 2. Parameter plot for NaZn13, showing the line corresponding to a regular icosahedron, the point corresponding to a regular snub cube, and the parameters actually obtained in the previous and present investigations.

In Fig. 2 conditions (11) and (12) are plotted on a parameter map together with the parameter values reported for NaZn13 by Zintl Hauke, those reported for KCd13 by Ketelaar, and those determined for NaZn13 in the present investigation. The uncertainties are indicated by the radii of circles drawn around the points determined by the parameter values since Zintl Hauke reported no uncertainty value, the uncertainty reported by Ketelaar (0-003) was assumed. It is seen that the parameter values obtained in the present work lie between those of Zintl Hauke and those of Ketelaar, and that they differ considerably from the values required either by a regular icosahedron or by a regular snub cube. 

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 dodecahydrododecaborate anion, B H 2-, is termed unique with considerable justification. This ion and its perhalo derivatives, e.g., Bi2C1i22-, are the most symmetrical molecular aggregates known. The boron atoms occupy the vertices of a regular icosahedron and each is bonded terminally to a hydrogen atom all boron atoms are environmentally equivalent.7,8 This anion is the only known example of the 7 symmetry group.8 General spectral, physical, and chemical properties of Bx2Hi22-are detailed in a paper by Muetterties et al.9... 

The molecular structure of the closo 12-vertex 1-SBnHn cluster (17) has been studied by electron diffraction methods augmented by ab initio calculations. Substantial distortions away from the regular icosahedron occurred by expansion of the pentagonal belt adjacent to sulfur.73 The UV-PES spectra of 17 has also been reported,71 and its microwave spectrum has been investigated and demonstrated that the molecule had C5v symmetry.74... 

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

Coordination 12 (regular, or approximately regular, icosahedron) 12 vertices (12 five-fold vertices) and 20 faces. 

Ih ,12(75,12(71,20C 3,15C 2, fl2S io,12S o, 2056,15(7 in. Continuous groups symmetry groups of linear molecules some viruses regular icosahedron... 

Fig. 10-5.—The atomic arrangement in e molecule of decaborane, Bi0Hu. Note that the 10 boron atoms are approximately at corners of a regular icosahedron. Each of the two remaining corners of the icosahedron may be considered to have been replaced by two bridging hydrogen atoms. The other 12 hydrogen atoms are bonded to boron atoms in such a way that the B—H bonds extend out from the center of the icosahedron.

It is possible to retain ligancy 12 with a central sphere as much as 10 percent smaller than the surrounding spheres. These spheres are then arranged at the corners of a regular icosahedron, which has 20 triangular faces (Figure 10-1). 

The [BI2H 2]2- ion is a regular icosahedron of atoms, each of the twenty faces being an equilateral triangle (Fig. 16.45a). All of the hydrogen atoms are external to the boron icosahedron and are attached by terminal B—H bonds. The icosahedron itself involves a resonance hybrid of several canonical forms of the type shown in Ffg. 6.45b and c. Both two-electron, two-center B—B and two-electron, three-center B—B—B bonding are involved. 

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 1-16 An idealized view of the structure of Fe3(CO,2), where the set of CO groups is represented as a regular icosahedron and the two orientations for the Fe3 cluster are shown.

The involvement of qrmmetry in chemistry has a long history in 640 B.O. the Society of Pythagoras held that earth had been produced from the reguliu hexahedron or cube, fire from the r ular tetrahedron, air fiom the regular octahedron, water from the regular icosahedron, and the heavenly sphere from the regular dodecahedron. Today, the chemist intuitively uses symmetry every time he recognizes which atoms in a molecule are equivalent, for example in pyrene it is easy... 

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