A microscopic mathematical analysis of diamond suggests that it has but one chemical cousin dubbed by Japanese mathematicians the "K₄ crystal". There could, it seems, only ever be one structure related to diamond, sharing its hereditary and symmetry properties, but with one unique property - chirality. One aspect of the discovery highlights the gulf that now exists between some sciences and mathematics despite efforts of funding agencies and other bodies to promoted joined up research.

Diamonds have entranced humanity for centuries, it is their fire, their unequalled hardness, and perhaps most of all their uniqueness that makes them the best friend of Marilyn Monroe clones everywhere. But, a Japanese mathematician with a keen interest in complex analytic geometry, spectral geometry, dynamical systems, and graph theory, has demonstrated that, while diamond is not quite as unique as we thought, it is not a common structural form either, being limited to just two crystal types.

Toshikazu Sunada of the Meiji Universtity explains in the February issue of Notices of the American Mathematical Society that at least some aspects of diamond's beauty lie in its crystal structure. This structure has rather special symmetry properties not seen in any other known crystal. However, Sunada's research has demonstrated that of an infinite universe of mathematical crystals there is one other that shares precisely the same structural properties as diamond, a structure Sunada calls the "K₄ crystal".

Sunada concedes that he thought his discovery of the K₄ crystal was unique, but following a flurry of media attention triggered by an AMS press release, Sunada was deluged with emails from chemists and crystallographers pointing out that this crystal structure is well known. "A few people  pointed out this fact," Sunada told SpectroscopyNOW, "They were rather sympathetic to that the difference  of culture between mathematics and the other sciences that led to the oversight, but the correspondents were also unaware of my work related to the present article which appeared in  several jornals over the last decade."

Nevertheless, Sunada's mathematical analysis idealises diamond as being composed of point carbon atoms, at the vertices of the ideal bonds between them, as you would expect. The network, or graph, can be built up in Sunada's model by constructing the unit cell and then repeating it in periodic fashion. A process not unfamiliar to crystal modellers everywhere. He points out that such models are underpinned by two patterns: the pattern of edges connecting vertices in the building-block graphs (that is, the pattern of bonding relations between the atoms), and the periodic pattern joining the copies of the graphs. Again, nothing too unfamiliar in that. Such a model can be used to create an infinity of hypothetical mathematical crystals simply by tweaking the graphs and how they are joined. One might describe it as reverse engineering crystal structures.

In such a model, diamond crystals are revealed as having two key properties that apparently make them unique. They have "maximal symmetry". No structural deformation of its periodic arrangement will make the structure any more symmetrical than it is in nature. However, most crystal structures can be deformed into a crystal with maximal symmetry. But, diamond has a second characteristic, a "strong isotropic property". "For any two edges with the same  end point, there is a rotation or reflection preserving the crystal and  exchanging these two edges while other edges (with the same end point) are  fixed," explains Sunada. "Ordinarily, 'isotropic property' would describe there being no  distinction in any direction. The strong isotropic property is the  strongest one among all possible meanings of isotropy."

Sunada's modelling reiterates what chemists already knew, that diamond is less than unique in these two properties. However, it turns out that of all the crystal stuctures that might be constructed mathematically, there is but only one other that has maximal symmetry and strong isotropy like diamond.

One intriguing feature of the K₄ crystal that distinguishes it from the structure of diamond, is that it is chiral, or handed. Apparently, the K₄ structure occurs as the silicon atom arrangement in  strontium(II) silicide and isostructural compounds, Sunada adds.

The first description of the structure goes back to a pioneer of the area, A F Wells, who called  it (10,3)-a [A. F. Wells, "Three Dimensional Nets and Polyhedra", Wiley  (1977)]

"The K₄ crystal looks no less beautiful than the diamond crystal," Sunada says, "Its artistic structure has intrigued me for some time." At the time of writing his Notices of the AMS paper, Sunada concedes he did not know whether or not the structure exists in nature or whether it had been synthesised. "Even though it turns out to be known to  crystallography, my result pinning down that  there are only two crystals having these properties is new," he told us. Perhaps there are countless other discoveries waiting to be made that would only need two disparate sciences to be joined up to allow them to happen.

Related links:

• Notices Am Math Soc, 2008, 55, 208-215 (pdf file)
• Sunada Page

Article by David Bradley