Two-Dimensional Self-Trapping Structures in Three-Dimensional Space

Manturov, V.O.; Kanel-Belov, A.Y.; Kim, S.; Nilov, F.K.

Авторы: Manturov V.O., Kanel-Belov A.Ya, Kim S., Nilov F.K.
Журнал: Doklady Mathematics, Pleiades Publishing,
Год издания: 2024
Аннотация: It is known that a finite set of convex figures on the plane with disjoint interiors has at least oneoutermost figure, i.e., one that can be continuously moved “to infinity” (outside a large circle containing theother figures), while the other figures are left stationary and their interiors are not crossed during the movement. It has been discovered that, in three-dimensional space, there exists a phenomenon of self-trappingstructures. A self-trapping structure is a finite (or infinite) set of convex bodies with non-intersecting interiors, such that if all but one body are fixed, that body cannot be “carried to infinity.” Since ancient times,existing structures have been based on the consideration of layers made of cubes, tetrahedra, and octahedra,as well as their variations. In this work, we consider a fundamentally new phenomenon of two-dimensionalself-trapping structures: a set of two-dimensional polygons in three-dimensional space, where each polygonaltile cannot be carried to infinity. Thin tiles are used to assemble self-trapping decahedra, from which secondorder structures are then formed. In particular, a construction of a column composed of decahedra is presented, which is stable when we fix two outermost decahedra, rather than the entire boundary of the layer, asin previously investigated structures.
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