ABSTRACT
The very possibility of explaining orderings in nature by means of a hierarchical geometrical construction based on the systematic use of Platonic solids can be probably traced back to Johannes Kepler (1571-1630) who tried to explain the relative size of orbits of the six planets known at the time by successively inscribing an octahedron, icosahedron, dodecahedron, tetrahedron, and hexahedron of proper sizes in a series of concentric spheres. This cosmological model was published in his book Mysterium Cosmographicum (1597). Since then, the basic idea – introducing geometrical constraints in the orderings of matter –has reappeared in several scienti…c domains. In particular, the possible existence of materials containing units of icosahedral symmetry has received some attention in a variety of di¤erent contexts. Chemists became interested in icosahedra because several clusters of atoms
have been found to be closely related to the icosahedron point group symmetry, the celebrated case of fullerene being a paramount example of this. Another instance is provided by icosahedral arrangements of 13 atoms, which are common in gas phase metal clusters or the structures of many boron-rich solids containing icosahedral arrangements of boron atoms,[9, 10] sometimes
in
in terms of an icosahedron of icosahedra building in Section 1.2). in solid state physics, the interest on the possible role of
units in some condensed phases can be traced back at when the icosahedral thought came up in several contexts. known structures of intermetallic compounds that involve These structures are usually complex, with 20, 52, even more atoms in the unit cell. Relatively simple cubic icosahedral building blocks are provided by MoAl12, WAl12; In their structures, at each lattice point of a body-centered is a regular icosahedron of twelve aluminum atoms about metal atom which occupies the central position.[11] and co-workers successfully described a complex to (Al,Zn)49Mn32 alloy containing 162 atoms in its unit The structure is based on a body-centered lattice. At each is a small atom (Zn, Al) which is surrounded by an icosahedron of twelve atoms. This group is then surrounded by 20 atoms, at the corners of a pentagonal dodecahedron, each atom lying directly out from the center of one of the centers of the pentagonal faces of the icosahedron. Twelve more atoms lie out from the centers of the pentagonal faces of the dodecahedron. At this stage, the resulting cluster is composed of 45 atoms, the outer 32 of which lie at the corners of a rhombic triacontahedron, a polyhedron with 30 rhombic faces which can be obtained as the union of an icosahedron and a dodecahedron. The next shell consists of 60 atoms, each directly above the center of a triangle that forms one-half of each of the 30 rhombic faces of the underlying triacontahedron (Fig.2.2). These 60 atoms then lie at the corners of a truncated icosahedron, which has 20 hexagonal faces and 12 pentagonal faces. Finally, twelve additional atoms are located out from the center of 12 of the 20 hexagonal faces completing the impressive series of closed shell atomic clusters based on icosahedral symmetry. Following a di¤erent line of thought John D. Bernal (1901-1971) considered
the icosahedral coordination as a key to understand the structure of liquid water, precisely on the basis that objects of icosahedral symmetry cannot …ll three dimensional space, which naturally prevents periodic crystallization. In fact, J. D. Bernal was extremely keen on hierarchy as a principle of building things and generalizing crystallography,[14] and this guiding principle inspired Alan L. Mackay to construct an atomic cluster based on an arrangement of three successive icosahedral symmetry shells (Fig.2.3): an inner icosahedron, a double-sized icosahedron, and an icosidodecahedron. The atomic cluster introduced by Mackay has played a relevant role in the …eld of quasicrystal research. In fact, the …rst thermodynamically stable quasicrystal found exhibits a triacontahedral growth habit (see Fig.2.10 in Section 2.3.4) and a slightly modi…ed version of the Mackay icosahedron plays a fundamental role in some structural models proposed for icosahedral quasicrystals (see Section 3.2.6).
