There is a well-known Olympiad problem: if there are coins (convex figures) on a flat table, then one of them can be pulled off the table without touching the rest. For a long time, mathematicians tried to prove a spatial analogue of this statement, until a counterexample was constructed! The system of self-wedging cubes was discovered by A. Ya. Belov only in 2002.
An idea came up: there is often no crack in a small grain, a crack does not grow beyond the grain boundary, but the grains hold each other. This idea theoretically makes it possible to create composites in which cracks do not grow, in particular, ceramic armor. A crack does not have time to develop in a small grain, and its growth stops when it reaches the border. At the same time, there are locations of convex bodies (in particular, regular polyhedra) that support each other. This circumstance can make it possible to create composite materials that can withstand high pressures. These considerations are already being used in the creation of new materials (a mega-grant was won), in particular, .
The report is devoted to the theory of self-wedging structures and the recent breakthrough made by V. O. Manturov:
a) The existence of two-dimensional self-wedging structures in three-dimensional space.
b) Construction of self-wedging structures that are motionless when fixing two polygons.
c) The fundamental innovation of V. O. Manturov's last approach is that all structures of this kind can be composed of "infinitely thin" layers 鈥� polygons. Further work on self-wedging structures and their engineering applications is envisaged.
At the end of the lecture, several problems will be proposed, both purely mathematical and related to specific applications.
