Folding Polyominoes into (Poly)Cubes

Abstract

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability.

Folding polyiamonds into octahedra

“…Decision questions for folding (unit) cubes are studied by Aichholzer et al [2,3]. The half-grid model allows folds of all degrees along the grid lines, the diagonals, as well as along the horizontal and vertical halving lines of the squares.…”

“…The half-grid model allows folds of all degrees along the grid lines, the diagonals, as well as along the horizontal and vertical halving lines of the squares. In this model, every polyomino of size at least 10 folds into the cube [3]. The remaining polyominoes of smaller size are explored by Czajkowski et al [8].…”

Self-assembly of 3-D structures using 2-D folding tiles

“…Those in (b) are not compatible. 1 We refer to the vectors {(1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1), (0, 0, −1)}) by the shorthand notation {+x, −x, +y, −y, +z, −z} throughout this paper. 2 Non-overlapping placements refer to different tile locations.…”

“…In the enumeration in Figure 4, the polycubes in the blue squares actually don't have a vertex in the center. The vertices in the polycubes in red (1,2,5,9) have three edges and three faces incident on the center point, creating what we refer to as a simple vertex. Of these 4 vertices, 1 and 9 are the same vertex type, which we will refer to as a convex vertex, and 2 and 5 are the same vertex type, which we will refer to as a concave vertex.…”

“…In contrast to the aTAM, in the FTAM each glue type can be specified to either form rigid bonds (which force two adjacent tiles bound by such a glue to remain fixed in co-planar positions) or flexible bonds (which allow two adjacent tiles bound by such a glue to possibly alternate between being in any of three relative orientations, as shown in Figure 1). Because the FTAM is meant to be a test bed for flexible, reconfigurable self-assembling systems, we present a version of the model which makes many simplifying assumptions about allowable positions of tiles and dynamics of the self-assembly process, but which also differs greatly from previously studied self-assembling systems which allow reconfigurability [1,2,6,[8][9][10][11][12]16], other computational studies of folding such as [1,2], and algorithmic studies focused on constructing more simple 3D structures such as [13]. In Section 2, we formally introduce the FTAM and provide definitions and algorithms describing its dynamics.…”