TITLE: What is magical about Cartan's magic formula?
QUESTION [40 upvotes]: Why is Cartan's magic formula
$$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$
called "magic"?
Should it be considered a highly surprising result?  Does it "magically" prove several other theorems?  What is the etymology? (Why it is variously referred to as E.Cartan's formula and H.Cartan's formula?)

REPLY [18 votes]: I would have liked to put this as a comment but I do not have enough points to do it.
I think there is some magic in this formula because it tells us that the Lie derivative $\mathscr{L}_X$ is homotopic to zero with the homotopy $i_X$ going from top-right $\Omega^{p+1}(M)$ to bottom-left $\Omega^p(M)$ diagonally in the following diagram:
$$\require{AMScd}
\begin{CD}
\cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots \\
\qquad @V{\mathscr{L}_X}V{0}V @V{\mathscr{L}_X}V{0}V\\
\cdots @>{d}>> \Omega^p(M) @>{d}>> \Omega^{p+1}(M) @>{d}>> \cdots
\end{CD}$$