# Cutting Space into Regions with Four Planes

Cutting Space into Regions with Four Planes

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.