ИСТИНА

Consider a set of contacting convex figures in R^2. It is not difficult to prove that one of these figures can be moved out of the set by translation without disturbing others. Therefore, any set of planar figures can be disassembled by moving all figures one b one (as opposed to simple homothetic motion of the plane, which automatically puts the figures out of contact with each other). One may think that this statement holds for higher dimensions as well. However, attempts to generalize it to R^3 have been unsuccessful. And recently quite unexpectedly interlocking structures of convex bodies were found such that every body was locked! These structures cannot be disassembled by removing individual bodies one by one. The author proposed a follows mechanical use of this effect. In a small grain there is no room for cracks, and crack propagation should be arrested on the boundary of the grain. On the other hand, grains keep each other. So it is possible to get "materials without crack propagation" and get new use of sparse materials, say ceramics. Quite unexpectedly, such structures can be assembled with any type of platonic polyhedra, and they have a geometric beauty. The talk will be devoted to such structures. They may be interesting in nanotechnology. 1. Gal'perin, G.A., 1985, Russian Math. Surveys, 40, 229. 2. Kanel, A. Ya., 2001, A story of an olympiad problem, Matematicheskoe prosveshchenie, 3, No. 5, 207-208 (in Russian). Dyskin, A.V., Y. Estrin, A.J. Kanel-Belov & E. Pasternak, 2001. A new concept in design of materials and structures: Assemblies of interlocked tetrahedron-shaped elements. Scripta Materialia, 44, 2689-2694. Estrin, Y., A.V. Dyskin, E Pasternak, H.C. Khor & A J Kanel-Belov 2003. Topological interlocking of protective tiles for Space Shuttle. Phil. Mag. Letters, 83, 351-355. Dyskin, A.V., Estrin, Y., Kanel-Belov, A.J. and Pasternak, E. 2003. Topological interlocking of platonic solids: A way to new materials and structures, Phil. Mag. Letters Vol. 83, No. 3, 197-203.