Multidimensional case

"Multi" in this book means "more than two." Hence, the three-dimensional case covers part of the multidimensional case. Let's extend that to even more dimensions. Visualizing such a case isn't easy. It requires a leap of imagination. Also, we have to assume that the same principles that apply to the two- and threedimensional cases also apply to higher dimensions. (Actually, we just define the higher-dimensional case that way.) Therefore, the distance L between any two points in multidimensional space is

......(1)

where w is the last variable of the group.
Equation 1 applies to any number of dimensions. It's quite common in chaos analysis, including vector analysis. Some authors call it the "distance formula."
An important case in chaos analysis is that for which the first point is at the origin of a set of graphical axes. In that case, x₁, y₁, and so on, are all zero. The distance formula (Eq. 1) then reduces to: