Archimedean polyhedra

The choice of the alternative pairs of the Ih regular orbit vertices as in the second column of Figure 2.21, leads to a second Archimedean polyhedron spanning the Oeo orbit. This is small rhombicosidodecahedron. 

Archimedean polyhedron A polyhedron whose vertices are all equivalent and whose faces are all regular polygons of more than one type. 

Catalan polyhedron A polyhedron that has a duality relationship with an Archimedean polyhedron, with all faces of identical irregular polygons and more than one set of vertices. 

Generalized Archimedean polyhedron A polyhedron derived from an Archimedean solid by replacement of one set of regular faces by nonregular ones (e.g., squares by rectangles), but which has the fiiU symmetry of the parent Archimedean figure. 

Classical stmctures inelude the regular and Arehimedean polyhedra Archimedean meaning only n, the polygonality, is fractional. In the 19 century, Catalan identified the reeiproeal polyhedra to the Archimedean polyhedra where n and p were exehanged with each other. Catalan polyhedra can be constmcted from their reeiproeals by joining the midpoints of each face of the Archimedean polyhedron to each other [21]. This reciprocal nature occurs also in two dimensions, one can easily visualize the reciprocals of (4,4), (3, 6) and (6,3), for example but is apparently absent in 3-dimensional nets. 

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). 

Buckminsterfullerene, an allotrope of carbon, is topologically equivalent to a truncated icosahedron, an Archimedean solid that possesses 12 pentagons and 20 hexagons (Fig. 9-16). [17] Each carbon atom of this fullerene corresponds to a vertex of the polyhedron. As a result, C6o is held together by 90 covalent bonds, the number of edges of the solid. 

Figure 2.4 The 13 Archimedean polyhedra, which can be constructed from the Platonic solids by relaxation of the requirement that all polygonal faces of the polyhedron be equivalent. Archimedean polyhedra have equivalent vertex figures on all vertices. All faces remain planar and equilateral, but are of two or three distinct kinds.

A more unified perspective can be developed by starting with the polyhedra derived from the regular orbits in the two parent groups, cubic (Oh) and icosahedral (Ih). The various Archimedean and Platonic solids then follow by a process of collapsing vertices of these regular-orbit polyhedra. Each polyhedron appears as the realization of an orbit of Oh/Ih or a subgroup. 

Three or more molecules may assemble in the solid state to form a finite assembly with connecting forces propagated in 3D. The components of such an assembly will typically form a polyhedral shell. The shell may accommodate chemical species as guests. The polyhedron may be based on a prism or antiprism, as well as one of the five Platonic (e.g. cube, tetrahedron) or 13 Archimedean (e.g. truncated tetrahedron) solids.4... 

As stated, a finite assembly with components arranged in 3D will possess a structure that conforms to a polyhedron. The simplest polyhedra are the prisms while poly-hedra of increasing complexity include Platonic and Archimedean solids. 

Polyhedra related to the pentagonal dodecahedron and icosahedron In equation (1) for 3-connected polyhedra (p. 62) the coefficient of is zero, suggesting that polyhedra might be formed from simpler 3-connected polyhedra by adding any arbitrary number of 6-gon faces. Although such polyhedra would be consistent with equation (1) it does not follow that it is possible to construct them. The fact that a set of faces is consistent with one of the equations derived from Euler s relation does not necessarily mean that the corresponding convex polyhedron can be made. Three of the Archimedean solids are related in this way to three of the regular solids ... 

Consultation of polyhedron models revealed the structure of 1 to conform to a snub cube, one of the 13 Archimedean solids, in which the vertices of the square faces correspond to the comers of 2 and the centroids of the eight triangles that adjoin three squares correspond to the eight water molecules. Indeed, to us, the ability of six resorcin[4]arenes to self-assemble to form 1 was reminiscent of spherical viruses in which identical copies of proteins self-assemble, by way of noncovalent forces, to form viral capsids having icosahedral symmetry and a shell-like enclosure. In fact, owing to the fit displayed by its components, 1 exhibits a topology that agrees with the theory of vims shell stmcture which states that... 

Crystalline fluorocarbonates, Ba2R(C03)2F3 (R = Y, Gd), of interest fortheir potential application in optics, were synthesized hydrothermally from RFj and BaC03 in molten Ba(N03)2 at 700°C and 216 MPa pressure (Mercier and Leblanc 1991). In these fluorocarbonates the Ba and R cations occupy the center of distorted BaOgF4 and ROgp3 Archimedean antiprisms monocapped on a square face. Two Ba-containing polyhedra are connected by an R-containing polyhedron to yield Ba2ROi2p9 blocks which are then connected to form infinite two-dimensional slabs. 

The duals of the Archimedean solids are the Catalan polyhedra. They are named after the French-Belgian mathematician Eugene Charles Catalan, who first described them in 1862. Given the characteristics of their duals, discussed above, Catalan polyhedra have more than one type of vertices but all their faces are equivalent, noimegular polygons. Furthermore, each Catalan polyhedron has the same nnm-ber of edges and the same symmetry as its Archimedean dual. The names of the Catalan solids and their duals are given in Table 4. 

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