论文标题
将polyiamonds折叠成八面体
Folding Polyiamonds into Octahedra
论文作者
论文摘要
我们研究了折叠成最小但未研究的柏拉图固体的聚原木（由三角形网格产生的多边形） - 八面体。我们显示了许多结果。首先，我们表征包含正区域孔的可折叠聚原石，即每个孔都可以折叠。其次，我们表明，当凸polyiamond折叠到八面体时，仅当它包含五个polyiamonds之一时。我们第三提出一个尖锐的尺寸结合：虽然存在14号尺寸的可展开的聚原木，但每个大小的聚座座至少15倍。这显然意味着可以在多项式时间内测试给定的聚原木是否折叠到八面体中。最后，我们表明，对于任何正整数分配到面部，存在一个折叠到八面体的polyiamond，使得覆盖面的三角形数量等于分配的数字。
We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid -- the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exist a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.
