Research On The Common Developments Of Plural Cuboids

In this paper, we study common developments that can fold into two or more incongruent orthogonal convex polyhedra, that is boxes. In searching for common developments that fold into two or more incongruent orthogonal boxes, we start from a relative simple task: finding two or more incongruent orthogonal boxes whose surface areas are equal but sizes are different.The smallest surface area that different orthogonal boxes can appear is 22 which admits to fold into two boxes of size 1×1×5 and 1×2×3. All common developments of these two boxes have been already known. The next smallest integer N such that surface area N can fold into two different boxes is 30. Matsui tried to list the common developments of two different boxes of sizes 1×1×7 and 1×3×3 of surface area 30, but failed[4]. We filled this gap and research further.By a new algorithm on a supercomputer, we first enumerate all common developments of boxes of size 1×1×7 and 1×3×3. Previous research stopped due to the memory overflow caused by too many developments generated. After improving the algorithm, we obtained the number of common developments of boxes of size 1×1×7 and 1×3×3, which is 1076.We had already known another polygon that can fold into boxes of size 1×1×7 and ?5*?5*?5, whose surface area is also 30. So next we try to find a polygon of surface area 30 that folds into above three different boxes. It is a great improvement from previous known one that the smallest surface area that folds into three different boxes is greater than 500. To achieve that we propose a new efficient algorithm which determines if a polygon P that can fold into two boxes of size 1×1×7 and 1×3×3 can also fold into the box of size ?5*?5*?5 as follows.Our new positional relationship algorithm checked whether each of 1076 common developments of boxes of size 1×1×7 and 1×3×3 can fold into the cube ?5*?5*?5. As a result, nine out of 1076 developments can fold into the cube of size ?5*?5*?5. While eight developments have only one way of folding into the cube of size ?5*?5*?5, the other one development has two ways of folding into the cube, for there is an angle of 26.6 degrees to the unit square's edge clockwise and anti-clockwise. That is, the last development can actually fold into 4 boxes: 1×1×7, 1×3×3 and 2 different ways of ?5*?5*?5.