Science

"I can hardly doubt that when we have some control of the arrangement of things on a small scale we will get an enormously greater range of possible properties that substances can have."

In every branch of science in which structured spatial objects have to be analyzed, essentially one and the same basic strategy is used: the object is mentally decomposed into elementary building blocks and then an attempt is made to explore the local matching rules according to which the building blocks are to be assembled to yield the object. Modern chemistry explores this strategy from Platonic solids to Primitive organisms. A class of inorganic compounds that is unmatched in terms of molecular and electronic structural versatility, reactivity, and relevance to analytical chemistry, catalysis, biology, medicine, geochemistry, material science, and topology, demands multi-, inter-, and trans-disciplinary research of complex systems.

Sphere is one of the simplest possible figures with high symmetry. It is the perfect form of heavenly bodies and to quote Copernicus, "...the spherical is the form of greatest volumetric capacity, best able to contain and circumscribe all else; and all the separated parts of the world - I mean the sun, the moon, and the stars - are observed to have spherical form; and all things tend to limit themselves under this form - as appears in drops of water and other liquids…." A discrete molecule with spherical shape - buckminsterfullerene - was first recognized in the cluster of 60 carbon atoms. Its spherical shape is due to its 20 hexagons - the usual shape associated with carbon - being interrupted by 12 pentagons, which provide the motif to close the sphere. The solid structure is technically called a truncated icosahedron, a form belonging to one of the thirteen Archimedian polyhedra. There are only five regular polyhedra known as Platonic solids because they constituted an important part of Plato's natural philosophy. Among these, the existence of the tetrahedron, the cube, and the octahedron is a fairly trivial geometric fact but the regular icosahedron and dodecahedron forms are nontrivial and more beautiful. The Archimedian polyhedra are obtained by symmetrically chopping off the corners of the Platonic solids ( for example, see Fig.1).

Fig. 1. Relationships between solids with icosahedral symmetry: from Platonic to Archimedian by different degrees of truncation

The dodecahedron has a pentagonal symmetry and it is believed that a conventional crystal can not have this symmetry. Many primitive organisms have the shape of pentagonal dodecahedron as their defense against crystallization. However, several "spherical" viruses possess approximately icosahedral symmetry.

The icosahedral 5: 3: 2 point group generates 60 "asymmetric units", the largest possible order of a point group. It has 12 five-fold, 20 three-fold and 30 two-fold axes. It is mathematically impossible to generate a sphere without discontinuities with any higher combination of symmetry elements.

Another way to describe the generation of a sphere is to make use of a theorem by Gauss, which states that to enclose a volume by folding a flat surface one must introduce wedge disclinations totalling 720⁰. This means simply a wedge cut from the surface at a point followed by a joining of the two sides of the wedge to form a cone at the point. The highest-order regular polygon that tiles a flat surface is a hexagon. Introducing 12-wedge disclinations of 60⁰ into a surface tiled by hexagons converts 12 hexagons into pentagons, leaving behind an integral number of equal wedges that total 720⁰. In buckminsterfullerene, made up of 60 carbon atoms, the association of 12 pentagons cause the spherical shape. However, the driving force to attain such a spherical structure by a carbon cluster is not clearly understood. Synthetic chemists were engaged in creating aesthetically pleasing spherical shaped molecules wherein they understood the importance of pentagon. A pentagon is available in pentagonal bipyramidal molecules. Early transition elements, especially of the second and third series, show rich hepta-coordination with the desired pentagonal bipyramidal symmetry. Thus the goal of current chemical research in this area is to generate nanosized molecular materials deliberately from hexagonal and pentagonal units with well defined properties. This type of unit-construction principle seems similar to that in the complex biological world . It is now possible to enter the `nanoworld'step by step by means of directed self-assembly processes.

Fig.2. KEPLERATE Mo72Fe30

In 1998, a cluster of the metallic element molybdenum and oxygen was made. It is a giant molecular sphere whose high molecular weight (~29000) is comparable to the molecular weight of many proteins. The sphere is made up of 132 molybdenum atoms and 72 of them constitute 12 pentagons (comprising pentagonal bipyramids) of six molybdenum aggregates. As this nano-sized molecule is diamagnetic, attempts were made to substitute paramagenetic centers retaining the spherical shape of the molecule.

The latest creation, reported in Angewandte Chemie (International edition, 1999, vol.38, p.3238) is also a sphere made of 72 diamagnetic molybdenum and 30 paramagnetic iron atoms (Fig.2); a Keplerate (to honour Kepler) molecular formula:

[Mo₇₂Fe₃₀O₂₅₂(CH₃COO)₁₂{Mo₂ O₇(H₂O)}₂{H₂ Mo₂O₈(H₂O)} (H₂O)₉₁].ca.150H₂ O; crystallizes in rhombo- hedral space group R3, a = 55.13 ; c 60.19 Å, V = 158439 Å³].

This species containing 30 paramagnetic iron centers (each iron with five unpaired electrons) constitutes the first example of a discrete high-nuclearity spin cluster, which will help in understanding the link between simple paramagnets and bulk magnets. Interestingly, this giant hetero-metal spherical cluster functions as a host to retain discrete dinuclear oxo-molybdenum molecule as a guest inside the sphere. This raises the possibility of using this molecule to trap other guest molecules by opening and shutting a part of the sphere. These giant molecules are roughly three times larger than buckminsterfullerene. Also unlike carbon clusters these are highly soluble in water and are stable in solution within a varied biological pH range. Use of these molecules in virology is also a possibility. Further, the self organization of these spherical clusters under simpler chemical conditions has opened a new dimension to explore the existing theories under the spherical environment. The higher congener of molybdenum, tungsten, has no known chemistry to show the formation of pentagon motif but this methodology of the induced molecule self-organization has led to the demonstration of a text book example of combinatorial chemistry where similar spherically shaped tungsten clusters are formed within seconds in solution.

It was Kepler who first attempted to attain a unified view of his work both in astronomy and what we call today crystallography. According to Kepler's pre-telescopic planetary model, the greatest distance of one planet from the sun stands in a fixed ratio to the least distance of the next outer planet from the sun. There were five ratios describing the distances of the six planets which were known to Kepler. These ratios were found to be close to the ratios of the radii of the spheres to those of the circumspheres of the five Platonic solids. Kepler's planetary images based on the five Platonic regular solids are beautiful but sadly untrue in the post-telescopic period. Three centuries after Kepler's first attempt, the principle of symmetry in solids was successfully addressed in the discovery of X-ray diffraction from crystals. A crystal is a particularly well ordered arrangement of atoms. The crystalline structure is periodic and the lattice has been taken as the definition of the crystalline state. In the lattice, identical `unit cells' - each containing precisely the same distribution of atoms - the building blocks of the crystal fit together regularly and periodically to fill space. The unit cell structure of most crystals is based on such Platonic solids as the cube, the tetrahedron and the octahedron. Many reputed physics texts state that the icosahedron with fivefold symmetry is of no physical interest as icosahedron can not serve as the unit cell of any conventional crystal. It is simple to understand that it is impossible to tile a plane with pentagons (fivefold symmetry). The discovery of quasicrystals in the mid-eighties - underlying structure with fivefold symmetry - seems to open a new kind of order, neither crystalline nor amorphous. This new order is explained with the mathematical theory of tiles and tiling. Langmuir's super-cooled liquid also has fivefold symmetry as recently encountered in the study of super-cooled lead. However, the new order of these materials is achieved by exotic operations, far from spontaneous self-assembly scheme of the oxo-metal Keplerates possessing (inscribed) icosahedron symmetry . What will these molecules deliver to us? We do not have any ready made answer. But it would be my worst dream if I see that Plato, Newton, Heisenberg, Lavoiseuer, or Kekule are spending their time in front of a patent office.