Are Rotations Exponential? Understanding the Concepts Behind Euler Identity

[Opus_723]

This isn't a homework question, but I'm having trouble understanding something about rotations conceptually. While reading about the Euler Identity online, I keep running into a few things that I can't wrap my head around and never come with an explanation.

Here are the concepts I can't understand:

A half-turn is the square of a quarter turn.
Rotations cannot be added, only multiplied.

Why is this? Intuitively, if I'm thinking about rotating by a quarter turn, then another quarter turn, I add them together to get a half turn. Instead, it appears I have to multiply the quarter turn by itself. I don't understand this.

pi equals 2(pi/2), not (pi^2)/4

Basically, I keep running across descriptions of rotations as exponential growth, whereas all I can see when looking at them is linear growth. How are rotations exponential?

 

[chiro]

Consider the rotation matrix R that does you're quarter turn.

In a nutshell what it is saying is that your half turn is represented by

R(R(X)) where X is your point assuming you are rotating the point about the origin.

Consider your quarter turn to be R x X = X(0)

Apply that same rotation R x X(0) applies the rotation to the result of the 1st rotation.

So essentially R^2(X) is applying the same rotation "twice" hence the half turn result.

If you want to prove this rigorously, create the rotation matrix R using the definition of your rotation and calculate R^2 and see what you get for each individual entry in your resultant matrix R^2.