Euler's formula seems a little anticlimactic after the mathematical preparation that paved the way for it. That's more common than not. Einstein's equation appears, a bit adrift, in a sea of mathematics on the page where it's derived; just as Newton, in the midst of expounding details of his Calculus, mentions the value of pi (to a record-breaking 17 decimal places) in a footnote, a mere fact in the progress of theory.

It's an appropriate disproportion. The key to the appreciation of e, i and π is that they transcend their symbols. They convey much more than a modest number like 2 because they represent far more than just an amount. They are each a system of ideas compressed into a single object that can simultaneously behave humbly like a number or intricately like machinery.

Euler's great work on analysis appeared in the 18th century. After all this work, we're still a few centuries behind the times, and mathematics has not meanwhile waited for us. The remaining unexplored regions justify a couple of footnotes.

A related quantity, elegant and enigmatic, is i raised to its own power: i ⁱ. It's so conceptually spare that it ought to evaluate to almost nothing at all. But its value can be determined easily from Euler's formula, and it's as unexpectedly complicated as e ^iπ was unexpectedly simple:

After writing this on his blackboard at Harvard, Benjamin Peirce is said to turned to his class and announced "Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important".

Finally, an obvious question to pursue is the mathematics of the square root of i. It should have its own pattern of behavior that can be derived much as that of i was. No one seems to have explored it yet; you could be the first.