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A crystal as beautiful as a diamond

Why are diamonds so shiny and beautiful? A Japanese mathematician says it's because of their unique crystal structure and two key properties, called 'maximal symmetry' and 'strong isotropic property.' According to the American Mathematical Society (AMS), he found that out of all the crystals that are possible to construct mathematically, just one shares these two properties with the diamond. So far, his K4 crystal exists only as a mathematical object. And nobody knows if it exists -- or if it can be synthesized. So will we say one day "A K4 Crystal Is Forever"? Read more...
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Written by Roland Piquepaille, Inactive on

Why are diamonds so shiny and beautiful? A Japanese mathematician says it's because of their unique crystal structure and two key properties, called 'maximal symmetry' and 'strong isotropic property.' According to the American Mathematical Society (AMS), he found that out of all the crystals that are possible to construct mathematically, just one shares these two properties with the diamond. So far, his K4 crystal exists only as a mathematical object. And nobody knows if it exists -- or if it can be synthesized. So will we say one day "A K4 Crystal Is Forever"? Read more...

A representation of the K4 crystal

You can see above an image of the K4 crystal (Credit: image created by Hisashi Naito, for Meiji University). This crystal was named the K4 crystal simply because it is made out of a graph called K4 -- in other words, a tetrahedron. The image above is based on the research work done by Professor Toshikazu Sunada, Meiji University, Tama-ku, Kawasaki, Japan.

The AMS news release gives more details about the two key properties of a diamond crystal. "The first, called 'maximal symmetry,' concerns the symmetry of the arrangement of the building-block graphs. Some arrangements have more symmetry than others, and if one starts with any given arrangement, one can deform it, while maintaining periodicity and the bonding relations between the atoms, to make it more symmetrical. For the diamond crystal, it turns out that no deformation of the periodic arrangement can make it any more symmetrical than it is. As Sunada puts it, the diamond crystal has maximal symmetry."

This property is not unique to diamonds. "Any crystal can be deformed into a crystal with maximal symmetry, so that property alone does not distinguish the diamond crystal. But the diamond crystal has a second special property, called "the strong isotropic property". This property resembles the rotational symmetry that characterizes the circle and the sphere: No matter how you rotate a circle or a sphere, it always looks the same. The diamond crystal has a similar property, in that the crystal looks the same when viewed from the direction of any edge. Rotate the diamond crystal from the direction of one edge to the direction of a different edge, and it will look the same."

So what did Sunada find? "It turns out that, out of all the crystals that are possible to construct mathematically, just one shares with the diamond these two properties. Sunada calls this the K4 crystal, because it is made out of a graph called K4, which consists of 4 points, in which any two vertices are connected by an edge."

For more information, this research work is featured in the February 2008 issue of the Notices of the American Mathematical Society under the title "Crystals That Nature Might Miss Creating" (Volume 55, Number 2, Pages 208-215, PDF format, 8 pages, 802 KB). The above image was extracted from this paper.

It's quite unusual that a researcher expresses his feelings about his work. But here is a short excerpt of what Sunada wrote. "The K4 crystal looks no less beautiful than the diamond crystal. Its artistic structure has intrigued me for some time. The reader may agree with my sentiments if he would produce a model by himself by using a chemical kit."

And here is one of his conclusions. "At present, the K4 crystal is purely a mathematical object. Because of its beauty, however, we are tempted to ask if the K4 crystal exists in nature, or if it is possible to synthesize the K4 crystal. More specifically, one may ask whether it is possible to synthesize it by using only carbon atoms."

And why not? The AMS news release adds some precisions. "The Fullerene, which has the structure of a soccer ball (technically called a truncated icosahedron), was identified as a mathematical object before it was found, in 1990, to occur in nature as the C_60 molecule."

Sources: American Mathematical Society news release, January 3, 2008; and various websites

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