Cutting Space into Regions with Four Planes

In the 1965 video Let Us Teach Guessing, George Pólya leads students to discover the maximum number of regions into which planes divide 3-space. (The problem of determining the maximum number of regions into which hyperplanes can divide -space is connected to recursion, induction, and binomial coefficients.)

In conducting a similar interactive activity, I have noticed that the first significant stumbling block for students is the case of four planes. Students have difficulty in visualizing configurations of four planes, and when students are able to visualize such configurations, they have trouble communicating their understanding to their peers. Thus I wanted to provide a Demonstration so students could experiment with moving a plane through space relative to the standard octants and/or share their discoveries about configurations of four planes in 3-space.

In particular, this Demonstration was built to accompany the forthcoming textbook Discrete Mathematics with Ducks, where the situation of cutting a yam arises in the chapter Counting and Geometry. Of course, anyone can enjoy exploring configurations of four planes in 3-space!