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An example of 3-D printed solids for the cube puzzle-a qiandu, a yangma, and a bie'nao, from left to right.
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( Download an STL file, suitable for direct 3-D printing of the three pieces of the cube puzzle.)įigure 19. About half way through the printing task, small magnets can be dropped inside each solid for easy assembly, if so desired. Once the three solids are tweaked to the students ' satisfaction, they can be exported to a STereoLithography (STL) file and sent to a 3-D printing station, where the STL files will be processed (sliced) and printed. This is a pleasant experience for the majority of students, who have rarely manipulated a virtual cube in such a manner. Once the cube has been split into its qiandu (triangular prism), yangma (rectangular pyramid), and bie'nao (triangular pyramid) pieces, students should be allowed to play with them within 123D Design and tweak their properties, such as colors and material, in order to develop a better understanding and appreciation of the puzzle and the geometric processes involved. The cube is now split into Liu Hui's cube puzzle consisting of a quiandu, a yangma, and a bie'nao. Use the split solid tool to dissect the chosen quiandu into a yangma (rectangular pyramid) and a bie'nao (triangular pyramid).įigure 18f. Sketch a diagonal polyline on one of the square faces of a qiandu.įigure 18e. The cube is now cut into two quiandu.įigure 18d. Use the split solid tool to cut the cube in half into two quiandu (triangular prisms).įigure 18c. Sketch a diagonal polyline on one of the faces in preparation to split the cube.įigure 18b. Perform the following steps, illustrated in Figures 18a-f, below, to carry out Liu Hui's cube dissection using 123D Design.įigure 18a. Liu Hui's Cube Puzzle can be readily built in 123D Design® using two splitting actions. Download a 123D Design file for Liu Hui's dissection of an arbitrary rectangular prism into a quiandu, yangma, and bie'nao.)įigure 18. ( Download a 123D Design file for Liu Hui's dissection of the cube into a quiandu, yangma, and bie'nao.
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Figures 18a-f, below, show how to apply the two splitting actions in 123D Design in order to perform Liu Hui's dissection of the cube. These splitting operations are accessible to middle and secondary students as well as to prospective teachers when using 123D Design. It turned out that Liu Hui's Cube Puzzle can be readily built out of a cube in 123D Design, using two splitting operations, both along a face diagonal. In addition, 123D Design has a simple and friendly user interface, which can be quickly introduced to middle and secondary students as well as prospective and classroom mathematics teachers, using specific design tasks.
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I chose to use Autodesk 123D Design®, which is free for educational purposes. It is beneficial for prospective mathematics teachers to see the connections between basic 3-D design and the mathematics they are learning and will be teaching to school children. As a matter of fact, the cube is a fundamental starting point for all 3-D design software. Therefore, when 3-D printing service became available at my university library, I decided to take one step further along this cube excursion and explore Liu Hui's Cube Puzzle as a 3-D design task.
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Meanwhile, 3-D design software has become freely accessible, inviting mathematics educators to engage students with mathematical applications and design-based learning. Although paper folding is practically adequate for classroom use, it is lacking in accuracy. However, GeoGebra did not yet support 3-D printing as of early 2017. GeoGebra 3-D modeling is pedagogically powerful, allowing students to develop imaginative skills for sense-making in a 3-D environment.
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