Interlocking structures

Consider a set of convex figures in R^2. It can be proven that one of these figures can be moved out of the set by translation without disturbing the others. Therefore, any set of planar figures can be disassembled by moving all figures one by one. However, attempts to generalize it to R^3 have been unsuccessful and finely quite unexpectedly interlocking structures of convex bodies were found. These structures can be used in engineering. 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. Surprisingly, such structures can be assembled with any type of platonic polyhedra, and they have a geometric beauty.Some pictures of interlocking structures can be seen in http://arxiv.org/abs/0812.5089.