Deza, Michel; Matematični kolokvij oktober 2010

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Space Fullerenes

Michel Deza

European Academy of Science, Vice President

7. oktober 2010

It is a joint work with Mathieu Dutour and Olaf Delgado.

A (geometric) fullerene is a 3-valent polyhedron whose faces are hexagons and pentagons (so, 12 of them). A fullerene is said to be Frank-Kasper if its hexagons are adjacent only to pentagons; there are four such fullerenes: with 20, 24, 26 and 28 vertices.

A space fullerene is a 4-valent 3-periodic tiling of \mathbb{R}^3 by Frank-Kasper fullerenes. Space fullerenes are interesting in Crystallography (metallic alloys, zeolites, clathrates) and in Discrete Geometry. 27 such physical structures, all realized by alloys, were known before.

A new computer enumeration method has been devised for enumerating the space fullerenes with a small fundamental domain under their translation groups: 84 structures with at most 20 fullerenes in the reduced unit cell (i.e., by a Biberbach group) are found. The 84 obtained structures have been compared with 27 physical ones and all known special constructions: by Frank-Kasper-Sullivan, Shoemaker-Shoemaker, Sadoc-Mossieri and Deza-Shtogrin. 13 obtained structures are among above 27, including A15, Z, C15 and 4 other Laves phases.

Moreover, there are 16 new proportions of 20-, 24-, 26-, 28-vertex fullerenes in the unit cell. 3 among them provide first conterexamples to a conjecture by Rivier-Aste, 1996, and to the old conjecture by Yarmolyuk-Kripyakevich, 1974, that the proportion should be a conic linear combination of proportions (1:3:0:0), (2:0:0:1), (3:2:2:0) of A15, C15, Z.

So, a new challenge to practical Crystallography and Chemistry is to check the existence of alloys, zeolites, or other compounds having one of 71 new geometrical structures.

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