Flex made adding controls to web content easy. So I did this mathematical illustration. You can drag the red “x” and fiddle with the number.

When I originally posted it, probably 2008 or so, I said, “Until I put up an explanation, I’m sorry to say that it is likely to be something of a riddle.”

Early in 2020, I said, “Well, here it is 2020 and I still haven’t put up the explanation. Until I do, I will give the hint that it connects Euler’s formula to the compound-interest formula for the exponential (see characterization #1 here).”

July 2020: I’m finally putting up some explanation!

One day, teaching calculus at MIT (this would have been the fall semester of 1984 or 1985), I had an extra class with nothing to teach for some reason, and I tried to think of a “treat.” Euler’s formula


is always high on the treat list, but the standard, very beautiful proof uses infinite series, which we wouldn’t cover until the spring. But it suddenly occurred to me that you could use the geometrical interpretation of complex multiplication–that multiplication by a complex number involves rotating and scaling–to prove the result from the compount interest formula


The proof involved a couple of applications of l’Hopital’s rule, which we had covered, so it was actually a good exercise.

I thought of this as something of a joke, since the infinite series proof is so beautiful and this one involves a fair amount of machinery, so I was surprised to see the same argument show up in a serious attempt to explain the formula (scroll down to “The Nitty Gritty Details” on that page if you are impatient).

Somewhere along the way, possibly while reading that post, it occurred to me that this is the intuitive way to think about the exp map for matrix groups: you are taking an element of the Lie algebra, which is an infinitesimal transformation, and applying a scaled-down version repeatedly to get an element of your Lie group.

So in the little Flex applet, you drag the red “x” up and down to change the value of \(\theta\) and you change the counter to change the value of \(n\). The applet shows you the repeated complex multiplication, and as \(n\) gets larger, you can see that the value approaches a point on the unit circle.