The 4-cube is the set of all points in __[-1,1__?]^4 .
The hyperplane { (x,y,z,w) : x + y + z + w = 0 } cuts the 4-cube
in the desired manner.

Now, { (.5,.5,-.5,-.5), (.5,-.5,.5,-.5), (.5,-.5,-.5,.5) } is an orthonormal basis for the hyperplane. Let (a,b,c) be a point on the hyperplane with respect to this basis. (a,b,c) is in the 4-cube if and only if |a| + |b| + |c| <= 2. The shape of the intersection is a regular octahedron.