Let's begin with orthogonal geometry, since that's what we're talking about here:
(n = 0)
In zero dimensions you have a point. Nothing more, nothing less. It has no area, no volume—it exists only as a dot of infinite smallness.
(n = 1, corners = 2, edges = 1)
If we leave that point and create a second point, we've got one dimension. For purposes of example, we'll call the length of the line "one", so
the two points have coordinates 0 and 1, with only one possible line connecting the two of them.
Points:
0
1
Edges:
0 - 1
(n = 2, corners = 4, edges = 4, planes = 1)
Making a parallel copy of that line distance one away from the original line at a right angle to the original, you get a square. Four points connected
by four edges and those four lines define a plane, face or square.
Points (Corners):
(0,0)
(0,1)
(1,0)
(1,1)
Edges:
(0,0) - (0,1)
(0,1) - (1,1)
(1,0) - (1,1)
(0,0) - (1,0)
Planes (Squares):
(0,0) (0,1) (1,0) (1,1)
(n = 3, corners = 8, edges = 12, planes = 6, cubes = 1)
Making a parallel copy of that square distance one away from the orignal square at a right angle to the original, you get a cube. 8 points connected
by 12 edges
Points (Corners):
(0,0,0)
(0,0,1)
(0,1,0)
(0,1,1)
(1,0,0)
(1,0,1)
(1,1,0)
(1,1,1)
Edges:
(0,0,0) - (0,0,1)
(0,0,0) - (0,1,0)
(0,0,0) - (1,0,0)
(0,1,1) - (0,0,1)
(0,1,1) - (0,1,0)
(0,1,1) - (1,1,1)
(1,1,0) - (1,1,1)
(1,1,0) - (1,0,0)
(1,1,0) - (0,1,0)
(1,0,1) - (1,1,1)
(1,0,1) - (1,0,0)
(1,0,1) - (0,0,1)
Planes (Squares):
(0,0,0) (0,0,1) (0,1,0) (0,1,1)
(0,0,0) (1,0,0) (1,1,0) (0,1,0)
(0,0,0) (1,0,0) (1,0,1) (0,0,1)
(0,1,0) (1,1,0) (1,1,1) (0,1,1)
(0,0,1) (1,0,1) (1,1,1) (0,1,1)
(1,0,0) (1,0,1) (1,1,0) (1,1,1)
Cubes:
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)
With me so far?
Let's take it to the 4th dimension!
(n = 4, corners = 16, edges = 32, planes = 24, cubes = 8, hypercubes = 1)
If we copy our cube and move it perpendicular in the 4th dimension to the existing cube, we get…
Points (Corners):
(0,0,0,0)
(0,0,0,1)
(0,0,1,0)
(0,0,1,1)
(0,1,0,0)
(0,1,0,1)
(0,1,1,0)
(0,1,1,1)
(1,0,0,0)
(1,0,0,1)
(1,0,1,0)
(1,0,1,1)
(1,1,0,0)
(1,1,0,1)
(1,1,1,0)
(1,1,1,1)
Edges:
(0,0,0,0) - (0,0,0,1)
(0,0,0,0) - (0,0,1,0)
(0,0,0,0) - (0,1,0,0)
(0,0,0,0) - (1,0,0,0)
(0,0,0,1) - (0,0,1,1)
(0,0,0,1) - (0,1,0,1)
(0,0,0,1) - (1,0,0,1)
(0,0,1,0) - (0,0,1,1)
(0,0,1,0) - (0,1,1,0)
(0,0,1,0) - (1,0,1,0)
(0,0,1,1) - (0,1,1,1)
(0,0,1,1) - (1,0,1,1)
(0,1,0,0) - (0,1,0,1)
(0,1,0,0) - (0,1,1,0)
(0,1,0,0) - (1,1,0,0)
(0,1,0,1) - (0,1,1,1)
(0,1,0,1) - (1,1,0,1)
(0,1,1,0) - (0,1,1,1)
(0,1,1,0) - (1,1,1,0)
(0,1,1,1) - (1,1,1,1)
(1,0,0,0) - (1,0,0,1)
(1,0,0,0) - (1,0,1,0)
(1,0,0,0) - (1,1,0,0)
(1,0,0,1) - (1,0,1,1)
(1,0,0,1) - (1,1,0,1)
(1,0,1,0) - (1,0,1,1)
(1,0,1,0) - (1,1,1,0)
(1,0,1,1) - (1,1,1,0)
(1,0,1,1) - (1,1,1,1)
(1,1,0,0) - (1,1,0,1)
(1,1,0,0) - (1,1,1,0)
(1,1,0,1) - (1,1,1,1)
(I'll leave the planes & cubes for you to work out, but it pretty much follows from the above...)