We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid—for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere. These geometric results offer a new way to build programmable matter that is substantially more efficient than what is possible with a square $N\times N$ sheet of material, which can fold into all polycubes only of surface area $O(N)$ and may stack $\mathrm{\Theta }(N^2)$ layers at one point. We also show how our strip foldings can be executed by a rigid motion without collisions (albeit assuming zero thickness), which is not possible in general with 2D sheet folding.

To achieve these results, we develop new approximation algorithms for milling the surface of a grid polyhedron, which simultaneously give a 2-approximation in tour length and an 8/3-approximation in the number of turns. Both length and turns consume area when folding a strip, so we build on past approximation algorithms for these two objectives from 2D milling.

我们提供了两种通用的铰链图案，它们使一条材料可以折叠到3D多维数据集网格上由单位正方形组成的任何连接表面（例如，任何多维数据集的表面）。折叠是有效的：对于在拓扑上等效于球体的目标表面，条带仅需要具有两倍的目标表面积，并且折叠最多可以在任何位置堆叠两层材料。这些几何结果提供了一种构建可编程物质的新方法，该方法比使用正方形的方法效率更高$ñ\times ñ$ 一片材料，只能折叠到表面积的所有多立方体中 $Ø（ñ）$ 并可能堆叠 $\mathrm{\Theta }（{ñ}^2）$一层。我们还展示了如何通过无碰撞的刚性运动（即使厚度为零）执行带钢折叠，这通常是2D纸张折叠无法实现的。

为了获得这些结果，我们开发了用于铣削网格多面体表面的新的近似算法，该算法同时给出行程长度的2近似值和匝数的8/3近似值。折叠带钢时，长度和匝数都会占用面积，因此我们基于过去的2D铣削目标的近似算法。