# Hypercube Edges

## David Akers

Regarding the number of sides of an n-dimensional hypercube, here is how
I consider it:
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.