Faces of a Hypercube
A 4-D hypercube has 3-D faces (cubes), 2-D faces (squares), 1-D edges, and 0-D vertices. This applet should help you systematically cycle through all the 3-D faces and help you find all the faces and edges in other dimensions as well.
There are 4 basis vectors that determine the edges of the hypercube. Any set of 3 of them define a cube face. --Move the vertical slider to see the 3-D faces. --How many 3-D faces are there? Any two basis vectors define a plane parallel to a set of 2-D faces. --How many ways can the basis vectors be paired? --How many 2-D faces are parallel to any one pairing of the basis vectors? --How many 2-D faces are there? Each basis vector defines a direction parallel to a set of 2-D edges. --How many 2-D edges are there?