> In what way is e^i*pi = -1 a metaphor for anything?
Its a metaphor for taking the unit length vector [1,0] represented by the complex number 1+0i and rotating it 180 degrees to -1+0i...
> Mathematics is funny because it is presented in
> "reverse", i.e. not the way it was derived.
Its usually presented in both ways in most curricula, sometimes depending on where you read about it or who teaches/tells you about it. Most mathematical books include historical contexts and non-formal accounts of the way results were derived, specially for classic and old results such as Euler's Formula. In most modern topics sometimes the historical context for a theorem is not easy to understand (i.e. discrete signal processing or optimal control) and is only briefly mentioned.
I appreciate what you're saying, but see my response below about semantics.
If you are calling it a metaphor, then aren't you calling ALL equations metaphors? That is doing violence to the meaning of the word "metaphor".
There is for sure a "relation" (or association) between the symbols e^i*pi = -1 and the picture of a unit vector on a complex plane. But that relation is not a metaphor.
You can prove that equality using pure analysis, or pure geometry, using appropriate definitions. The metaphor is the intuition that the two proofs are equivalent in an abstract sense.
