# 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?

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Contributed by: Sarah-Marie Belcastro (September 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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!

## Permanent Citation

"Cutting Space into Regions with Four Planes"

http://demonstrations.wolfram.com/CuttingSpaceIntoRegionsWithFourPlanes/

Wolfram Demonstrations Project

Published: September 20 2011