I begin with an example. To move from one dimension to two dimensions, we simply stretch a line into a square. The line is "duplicated" at a distance equal to its length parallel to the first line. The number of vertices is simply doubled. We note that the new number of sides is found by counting the new side, the old side, plus the two sides connecting the duplicated vertices. This gives us four. We can easily extend this to the case in which three dimensions are extended into four. Here we are 'duplicating' a cube instead of a line or a square. The number of vertices will be doubled. The number of sides will be 2*cube_sides + vertex_connections = 2*12+8 = 32. We now generalize to the following formulas for sides and vertices of an n-dimensional cube:
number_of_sides[n] = number_of_vertices[n-1] + 2*number_of_sides[n-1]
number_of_vertices[n] = 2*number_of_vertices[n-1]
Thus, a fifth-dimensional hypercube would have 16 + 2*32 = 80 sides, and 2*16 = 32 vertices.