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. [br]--Move the vertical slider to see the 3-D faces.[br]--How many 3-D faces are there?[br][br]Any two basis vectors define a plane parallel to a set of 2-D faces.[br]--How many ways can the basis vectors be paired?[br]--How many 2-D faces are parallel to any one pairing of the basis vectors?[br]--How many 2-D faces are there?[br][br]Each basis vector defines a direction parallel to a set of 2-D edges.[br]--How many 2-D edges are there?