# Exploring Liu Hui’s Cube Puzzle - In the Classroom: 3-D Modeling Using GeoGebra

Author(s):
Lingguo Bu (Southern Illinois University Carbondale)

GeoGebra also comes with a powerful 3-D panel, where Liu Hui's Cube Puzzle can be modeled on the basis of a cube by constructing three face diagonals (FHBDBE) and one internal diagonal (BH) as shown in Figure 17. To define the three solids in Liu Hui's dissection, one can use the prism tool for the qiandu (green triangular prism) piece and the pyramid tool for the yangma (red rectangular pyramid) and the bie'nao (purple triangular pyramid). By rotating the 3-D view, one can obtain a better picture of all three solids. GeoGebra further has a net tool that flattens a chosen polyhedron to a template on a plane. Modeling Liu Hui's Cube Puzzle in GeoGebra 3-D allows students to appreciate the relationships between 3-D geometric operations and 2-D templates and further develop increasingly complex spatial reasoning skills, which are strikingly lacking in traditional school mathematics.

Figure 17a. Liu Hui's Cube Puzzle can be modeled using construction tools in the 3-D panel of GeoGebra. Click then drag the image above to see the dissection from different perspectives.

Figure 17b. Use the slider to fold up the 2-D nets into the 3-D pieces of Liu Hui's Cube Puzzle; namely, a red qiandu, green yangma, and blue bie'nao (GeoGebra applet by Lee Stemkoski).

Figure 17c. Use the slider to "solve" Liu Hui's Cube Puzzle; that is, to fit the red qiandu, green yangma, and blue bie'nao together to form a cube. You may also move the three pieces into position by dragging them (GeoGebra applet by Lee Stemkoski).

Lingguo Bu (Southern Illinois University Carbondale), "Exploring Liu Hui’s Cube Puzzle - In the Classroom: 3-D Modeling Using GeoGebra," Convergence (February 2017)