The cube, common as it is in everyday life, provides extraordinary opportunities for students to ex- perience mathematics in multiple domains (e.g., Senechal, 1990). It has served as the foundation for numerous mathematical toys and puzzles. It is also the basic reference framework of navigation in 3D learning and design environments. There are numerous cube-based learning activities that are both aesthetically engaging and mathematically rich for K-16 students (Bu, 2017, 2019, 2021). There is, however, little discussion in the literature about cube twisting – a playful three-dimensional transfor- mation that opens the door to a variety of mathematical ideas, both geometrically and algebraically. It is particularly interesting to twist a cube with the assistance of GeoGebra 3D Graphics or similar modeling technologies. There are two major types of twisting we can perform on a cube. In the first case, a cube is twisted typically 90 degrees while allowing its faces to stretch as if they were made of elastic faces, which can be called an elastic or flexible twist. In the second case, a cube is twisted for a certain angle while retaining its volume.

In a thought experiment, we could cut the cube horizontally into a large number of thin square slices, which are then gradually rotated for a certain angle, such as 90 degrees, from the bottom to the top. It is evident that the twisted cube has the same volume as the original. Figure 1 shows two 3D-printed physical models with a clockwise and a counterclockwise twist of 90 degrees, respectively. The second type of twist can be called a volume-invariant twist and is the focus of the present article. We first look into the process of GeoGebra Simulation and then, using GeoGebra models as a scaffold, we explore and verify the algebraic structures around the twisted cube, including the twisted faces, the spiral curves, and the area of a twisted cube face. The ideas we unveil on the journey range from elementary to postsecondary mathematics. Readers should feel free to make stops wherever appropriate or enjoy the full picture of a twisted cube and extend the methods of inquiry to other problem situations.

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