Dimensions Defined without Coordinates??? Mathematicians, I'm sure, have solidly clear idea of dimension but not to me. So allow me to ask some questions for all who have a clear understanding of the concept of dimension. Can we define dimension without the use of so called multidimensional reference frames with their equal number of coordinates? 3 coordinates in Cartesian system. 4 coordinates in Einstein's relativities. 10 and 11 coordinates(?) in superstring and M-theory. What is the true relational correspondence between dimension and direction? How can direction be defined without a coordinate system or a reference frame? What is the physical meaning of angles (plane, solid and abstract phase angle)? Can direction be defined without angles?