I'm just beginning to learn topology, and there's something that I realize has been nagging at me since roughly the eight grade.
Why does the Euclidean distance in N-dimensional space involve a bunch of squaring and square-rooting?
I understand it, mind you.
- In 1-space the "distance" is $\sqrt{\Delta x^2}$ is which is just $\Delta x$.
- In two-space, you're doing the Pythagorean theorem.
In 3-space, I visualize it like this: you want to find the distance $\Delta x, \Delta y, \Delta z$ so you:
- let $a = \sqrt{\Delta x^2 + \Delta y^2}$, $a$ is the distance along the $x, y$ plane
- the total distance is $\sqrt{a^2 + \Delta z^2}$
- the above expands to $\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$
To generalize, faced with $n$ dimensions, you pick two, find the distance along those two dimensions, plop a point down there, and repeat (until you have only one dimension left, at which point you're done.)
However, I don't understand why distance in n-space requires repeated squaring. Shouldn't there be a distance formula that requires cubing and the cube-root in three-space, or quading (is that a thing) and quad-roots in four-space, and so on? Just in the interest of symmetry, it seems weird that squares get special treatment.
I know this is a very philosophical question, but is there a way to find Euclidean distances in n-space that involves taking the nth power and nth root instead of repeatedly projecting down a dimension?