Making meaning of mathematical “bridges”

I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form.

I'm a math noob, but I've seen (as we all have) a few examples of "connections" between different areas of math by way of an equality or some other relating device. An example is:

e^iπ + 1 = 0

These five constants serve different functions in a variety mathematical tasks. It seems to me that any constant is a "measurement" of some kind, which then begs the question "What is it a measurement of?"

Can you think of some places to start? If you have a "bridge" you like, post it.

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Personally I find the "full" version of the formula ($e^{i\theta} = \cos{\theta} + i\sin{\theta}$) of which this "identity" is a special case to be much more beautiful and revealing – crasic Nov 12 '10 at 9:36
It only "begs the question" (google for that phrase to find out what it really means!) because you arbitrarily decided that constants are "measurements" (including complex ones...). If you had decided instead that constants are icecream flavours, you could would be asking "what flavour is is $e$?"... – Mariano Suárez-Alvarez Nov 12 '10 at 17:16