Can "direction" be quantized? That is the main question of this thread. At the outset, we need to clarify (hopefully come to some sorts of agreement) the meaning of the word "direction" for two groups of people, (1) the mathematicians and (2) the physicists. I am considering myself belonging to the physics group. So I am welcoming people from the math group to supply inputs for insights on the concept of "direction." In the future I am going to give bits and pieces of ideas (from my own independent research) on the physical side.