Cutting Space into Regions with Four Planes

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What is the maximum number of regions you can get by dividing space with four planes?


The three coordinate planes are fixed, and by themselves form eight regions. Using the sliders, rotate and translate the yellow plane to form varying numbers of regions. (For those who know some vector geometry, the rotational sliders control the components of the normal vector to the yellow plane.) You can also rotate the entire bounding cube to view the situation from different angles.

A harder question is to find the maximum number of regions into which planes divide 3-space.


Contributed by: Sarah-Marie Belcastro (September 2011)
Open content licensed under CC BY-NC-SA



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!

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