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 finally,
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.