I learned about "i" (the imaginary number, not myself) from a retired math professor who was tutoring a young kid in a school i was working in. He was teaching the kid about cycles and negative roots and many of the things you're talking about. This guy had three pieces of paper and was teaching the kid by switching between the uses of some combination of the two papers. I was interested in his little tutelage session after he was done with the kid and the retired professor enjoyed talking to me.
So, after he explained some things and i was quick on the uptake (which is why he enjoyed talking to me), i asked if i could have the papers, he wasn't going to use them so he gave them to me. Over the weekend, i sat for about 45 min trying to figure out the relationship between them. After a bit of shuffling, i realized that "i" showed up on one plane only when it intersected the plane (duh). When i correlated all the planes, i realized that the path that "i" was drawing was sort of a spiral. So, basically, "i" hits our normal number plane only once in a while, while the rest of its plotting is on the z axis, hence the spiral and its relative 'non-existence' other than when it was on that other plane. [He was absolutely thrilled when i showed him this and asked a few times (once overtly and twice implicating) whether someone helped me with this insight]
Now what i'm not getting from Euler's identity theorem is the diagram next to it, which is a circle that is hemi-inscribed. I'd be very interested to see that same proof plotted on a 3-D surface, and i'd wager that though the diagram on wikipedia is 'right', but only from the perspective of 'above'. What i'm saying is that the proof isn't saying anything about 'identity' because the end plot of the point is further up the axis that we're standing on, and from our vantage point we can't see that the 'circle' is really being inscribed on a spiral, and the start and end points are not in the same position.
Now that i re-read this post, and looked at the website again i'll be even bolder. I'll assert that the starting point for the diagram is (1,1,0) and ending point is (-1,-1,0) and that the hemi-circle IS traveling through the plane Z, and that the "+1" is moving diagonally toward the (0,0,0) point. It's all about perspective man, all about perspective. (and hence relevant to Steve's original discussion about race and gender . . . and when can we throw class into the mix?).
After saying that, what's really got my undies in a bunch is that i originally expected the identity theorem to have something that leaves and arrives at the same point (which it doesn't). But what the theorem is really stating is the proportional relationship between the numbers. Is this a correct interpretation marty?
Also, if i remember correctly, this was the same formula that Lakoff dealt with in the end of his book "where mathematics comes from". I understood the book, but with no math teacher to help, i didn't get the grit of his point for this particular equation.
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