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

Abstract

It is known that a finite set of convex figures on the plane with disjoint interiors has at least one outermost figure, i.e., one that can be continuously moved “to infinity” (outside a large circle containing the other 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-trapping structures. 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-dimensional self-trapping structures: a set of two-dimensional polygons in three-dimensional space, where each polygonal tile cannot be carried to infinity. Thin tiles are used to assemble self-trapping decahedra, from which second-order 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, as in previously investigated structures.