Two-dimensional self-interlocking structures in three-space

Abstract: It is well known that if there exists a finite set of convex bodies on the plane with non-overlapping interiors, then there is at least one "extremal" one among them, i.e., some one which can be continuously "taken away to the infinity" (outside a large ball containing all other bodies). In 3-space a phenomenon of self-interlocking structures takes place. A self-interlocking structure is such a set of three-dimensional convex bodies with non-overlapping interiors that any infinitesimal move of any of them is possible only as a part of the move of all bodies as a solid body. Previously known self-interlocking structures are based on configurations of cut cubes, tetrahedra, and octahedra. In the present paper we discover a principally new phenomenon of 2-dimensional self-interlocking structures: a family of 2-dimensional polygons in 3-space where no infinitesimal move of any piece is possible. (Infinitely thin) tiles are used to create {\em decahedra}, which, in turn, used to create columns, which turn out to be stable when we fix some two extreme tiles. Seemingly, our work is the first appearance of the structure which is stable if we fix just two tiles (and not all but one). Two-dimensional self-interlocking structures naturally lead to three-dimensional structures possessing the same properties.