ИСТИНА

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! An idea came up: there is often no crack in a small grain, a crack does not grow beyond the grain boundary, but the cracks 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 when creating new materials (a mega-grant was won http://hybrid-nano-lab.misis.ru/index.php), in particular, body armor http://www.3ders.org/articles/20130829-arl- purdue-explore-3d-printing-to-fix-damaged-on-the-spot-in-combat-zones.html, It is surprising that the system of self-wedging cubes (see the picture) was discovered by A.Ya. Belov only in 2002. Three-dimensional space is arranged in a very non-trivial way and our three-dimensional intuition is not sufficiently developed. The talk is devoted to the theory of self-wedging structures, and a recent progress made by V.O. Manturov, which consists in the following: a) The existence of two-dimensional self-wedging structures in three-dimensional space b) Construction of interlocking structures that are fixed when two polygons are fixed. c) The innovation of VO Manturov's last approach is that all structures of this kind can be folded from "infinitely thin" layers - polygons. Further work on interlocking 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. http://kvant.mccme.ru/pdf/2009/2009-01.pdf Vassily O. Manturov, Alexei Kanel-Belov, Seongjeong Kim, Two-dimensional self-interlocking structures in three-space, 2021 (Published online), 21 pp., ArXiv: 2109.06426 Dyskin, A.V., Y. Estrin, A.J. Kanel-Belov and E. Pasternak, “Interlocking properties of buckyballs.”, Physics Letters A, 319 (2003), 373–378 [14] Djumas, L., Simon, G. P., Estrin, Y. et al. Deformation mechanics of non-planar topologically interlocked assemblies with structural hierarchy and varying geometry. Naure, Sci Rep 7, 11844 (2017). Https://doi.org/10.1038/s41598-017-12147-3 A. J. Kanel-Belov, A. V. Dyskin, Y. Estrin, E. Pasternak, I. A. Ivanov-Pogodaev, “Interlocking of convex polyhedra: towards a geometric theory of fragmented solids”, Mosc. Math. J., 10: 2, Pesin Aleksandr Moiseevich, Belov Aleksey Yakovlevich, Diskin Arkady Viktorovich, Tulupov Oleg Nikolaevich, Pustovoitov Denis Olegovich, Lokotunina Natalia Mikhailovna, Biryukova Olesya Dmitrievna, Method for producing layered bimetal steel-aluminum alloy, patent for invention, Patent number: RU 2756086 C1, Patent Office C1 : Russia Year of publication: 2021 Application number: 2021103956, Date of registration: 16.02.2021, Date of publication: 27.09.2021, Patent holders: Federal State Budgetary Educational Institution of Higher Education “Magnitogorsk State Technical University named after G.I. Nosov ", 2021 Pesin Alexander Moiseevich, Kharitonov Veniamin Alexandrovich, Tulupov Oleg Nikolaevich, Belov Alexey Yakovlevich, Diskin Arkady Viktorovich, Pustovoitov Denis Olegovich, Lokotunina Natalia Mikhailovna, Voloka, Patent number: RU 2759362 C1 Patent Office: Russia Year of publication: 20218 Application number: 20211 .2021 Publication date: 12.11.2021 Patent holders: Federal State Budgetary Educational Institution of Higher Education “Magnitogorsk State Technical University named after G.I. Nosov "INTERNATIONAL PATENT CLASSIFICATION: B21C 3/00 Working tool of drawing mills; combination of dies and mandrels in drawing mills, 2021 Pesin Aleksandr Moiseevich, Kharitonov Veniamin Aleksandrovich, Korchunov Aleksey Georgievich, Belov Aleksey Yakovlevich, Pasternak Elena, Pustovoitov Denis Olegovich, Pivovarova Ksenia Grigorievna, Compound Voloka, Patent for invention, Patent number: RU 2759179 C1 Patent office: Russia Publication year: : 2021110189 Date of registration: 12.04.2021 Date of publication: 09.11.2021 Patent holders: Federal State Budgetary Educational Institution of Higher Education “Magnitogorsk State Technical University named after G.I. Nosov ", 2021