There are three magic numbers in mathematics that stand out above all else. The first one is π, the relationship between a circle and it's diameter; there's e, the natural logarithm that permeates so many areas of mathematics; and i, the square root of -1, which is an impossible and unthinkable number (for no number multiplied by itself can ever be negative).
These three numbers, e, i and π, have nothing to do with each other. They stand remote in meaning and effect from each other, all three of them deriving from different areas of mathematics and with long and different histories behind them. And yet, there is an equation that ties them together in a most astounding symmetry. It is:
eiπ = -1
All of these magical numbers sort of, in their own special way, cancel each other out. Where there is no apparent symmetry, if you only combine them together like the final pieces of a huge puzzle, suddenly a remarkable symmetry appears: It's minus one.
I've often thought about this minus one, too. It, too, represents a form of mathematics that is unusual. You can't really measure -1. You can't go -1 yards, you can't measure -1 ounce of flour. It is a reflection of real numbers; negative, elusive, but not without its own agreeable symmetry and beauty.
Some people like to write the equation as
eiπ + 1 = 0
This strikes me as enormously masculine, for some reason. The numbers are all positive; it's all bright and sunny, on the positive side of things, and seems to say "hey, look what a nice, happy result we get if we tweak this equation a little. We've got i, e, π, 1 and 0, and a plus sign. It covers everything fundamental in mathematics."
But I like the first way better; it seems more fundamentally feminine, in a way; almost a bit of Yin and Yang in a sense. The feminine side is not too concerned about shaping things right by force; it's happy with accepting a minus one just on its own merits. It shows you that you don't have to mangle everything to get a nice, round result from it; that things can end up with a negative one and still be beautiful in its own special way.
To me, the minus one is like a nymph in a secluded, enchanted forest, gazing lovingly down into a clear, dark pool of water at her own beautiful reflection; something that can not be touched, smelled, or sensed - just lovingly gazed at and admired from afar. And this is the ultimate beauty I see in the equation above: Three fundamental numbers in universe combined into one elusive, mystical reflection.
And they say math is boring.