Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $pi$ are related?

I read a bit about this equation: $e^{i\pi}=-1$
For someone knowing high school maths this perplexes me. How are these three irrational numbers so seemingly smoothly related to one another? Can this be explained in a somewhat intuitive manner?
From my perspective it is hard to comprehend why these almost arbitrary looking irrational numbers have such a relationship to one another. I know the meanings and origins behind these constants.

Answer

Just think about it this way: $\pi$ is related to the circle, whose equation is $x^2+y^2=r^2$. Euler's constant e is related to the hyperbola, whose equation is $x^2-y^2=r^2$. In order to turn $y^2$ into $-y^2$ we need a substitution of the form $y\mapsto iy$.