Each of these unique columner forms has an odd number of half twists in the surface so that both sides (and both edges!) of the surface are connected to one another. In mathematics these surfaces are said to be non-orientable since it is not possible to define one side to be "top" and the other as the "bottom" as long as they are connected to one another. Thus anything on the surface is unable to become "oriented" relative to the side of the surface upon which it is located.