Can any gain be made from looking at the Pythagorean theorem in another way? The typical way to use the known values of the lengths of the sides of a right triangle and calculate the length of the hypotenuse. The orthogonal vectors are usually more useful to know than the distance between the ends. What if we assumed, and it is a rather large assumption, that the sides of the triangle were equal and 1 unit long. We we find that the length of the hypotenuse is root 2. So let's assume that if we find anything with a length of root 2 it must be a hypotenuse of 2 dimensions. If we find something of root 3 in length it must be a 'hypotenuse' of a cube with each side one unit long. Lengths are not constant, they change with velocity. What if we took the derivative and used the number of dimensions to find the velocity or vice versa: 


What is a dimension?

Jan. 20 2012