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.
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.
"The K4 crystal looks no less beautiful than the diamond crystal," Sunada writes. "Its artistic structure has intrigued me for some time." He notes that, although the K4 crystal presently exists only as a mathematical object, it is tempting to wonder whether it might occur in nature or could be synthesized. This is not so far-fetched as it may sound: 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 C60 molecule.
http://www.sciencedaily.com/releases/200…