Liu Hui Cube Puzzle/Dissection (Qiandu, Yangma, Bie'nao)

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The cube can be dissected in numerous ways. One widely recognized dissection decomposes a cube (or any rectangular prism) into its halves, thirds, and sixths using two simple face diagonal cuts. This concept dates back to around 263 AD when a renowned Chinese mathematician, Liu Hui, illustrated the idea in his commentaries on the Chinese Nine Chapters of Mathematical Arts. Liu Hui's dissection can be easily achieved in a modern 3D design environment such as Autodesk Fusion 360 or similar software packages. Take a cube, slice it along a face diagonal, and you will get two halves. Each half is called a "Qian Du," using its original Chinese name. Keep one half piece and slice the other along another face diagonal; you will obtain a square pyramid, which is one third of the cube and is called a "Yang Ma." The third solid is a tetrahedron, which is one sixth of the cube and is called a "Bie Nao." It makes logical sense that two halves (Qian Du) make up a cube and three Yang Ma make up a cube. The Bie Nao are unique in their properties. If you take two Bie Nao and create a mirror image of one of them, the two Bie Nao will form a Yang Ma. Therefore, if you take six Bie Nao and mirror three of them, you could create three Yang Ma and further a whole cube. Exploring the visual, spatial, and quantitative relationships among these pieces can be an enjoyable experience. For printing purposes, 0.2 mm is sufficient. If you choose 0% infill, you can leave some small magnets inside when the pieces are halfway through the printing process. Please increase the shell to 4 if the walls appear too thin. For more information, please refer to Bu, L. (2017). Exploring Liu Hui's Cube Puzzle: From Paper Folding to 3-D Design. MAA Convergence. Accessible at https://www.maa.org/press/periodicals/convergence/exploring-liu-hui-s-cube-puzzle-from-paper-folding-to-3-d-design