Imagine that you are a 2-dimensional being living in a 2D world. A flat person in a plane... or a "Flat Land", as some might call it. You can only perceive objects which lie in your 2D plane.

Suppose that some supreme being takes a 3D cube and lowers the cube into your plane, such that the top face and bottom face are parallel to the plane. You would be able to perceive a 2D cross-section of the cube. This cross section would be shaped like a square.

What if the supreme being starts rotating the cube about one or more axes? The cross section you observe might no longer be a square. It could be a rectangle, a triangle, or some other strange shape.

Now let's increase the dimensionality. Imagine you are a 3-dimensional being (you are) living in a 3D world (it is). You can only perceive objects which lie in your 3D volume.

Supose that some supreme being takes a 4D cube (a hypercube, if you will) and puts it into your volume. You would be able to perceive a 3D cross-section of the hypercube. If the hypercube is correctly aligned, you would see a normal 3D cube.

What if the supreme being starts rotating the hypercube about one or more axes? Well... some funky stuff can happen. The goal of this page is to show you how those cross sections would look.

Have fun!