In order to understand the power of Euler’s identity eiπ + 1 = 0, you need to unlearn some pretty fundamental things you were taught in school.
We are all taught that light and sound travels in waves, that alternating current is described by sine waves, as well as tides, water waves, guitar strings, bouncing springs, pendulums and anything other kind of regular periodic movement.
The problem with this mental model is that it misses the most important thing: None of these things are best described by something moving in a sine wave. Rather, they are best described by something moving in a circle. Deep down, they are all pretty much equivalent to tying a rock to a rope and swinging it over your head in a circle.
I know this seems counterintuitive. After all, we can actually see the bob of the pendulum move left and right. The movement doesn’t look anything like a circle. The problem is that the circle isn’t visible — but even so, it is essential.
In the case of a pendulum, one axis of the circle is displacement and the other axis is speed. Any time you see periodic movement, you are actually looking at something that holds energy by spinning in a circle. Unfortunately one or more of the dimensions of that circle is often invisible to the naked eye.
Once you absorb this idea, Euler’s identity takes on much more significance. It’s the key to understanding light, sound, vibrating strings, bouncing springs, and pretty much everything that sends information around the Universe.
This post is just crying for one of your nifty Java toys to visualize the concept.
Thinking of this, I went looking for the wonderful visualizations Jim Blinn did for the Mechanical Universe and Project Mathematics!. Sadly, they’re not on-line (well, YouTube has a Spanish dub of The M.U., but the encoding’s so terrible you can barely read the math).
Had the producers who worked with Blinn done these works in this century, they’d be up on the web for us all to enjoy, a la the Khan Academy. But his work remains trapped in 90s style video distribution (“to order DVDs, send a check to…”). Somebody needs to use the current cheer-leading of STEM education to fund republishing those shows on the web.
Khan’s lectures are great, but the clever methods Blinn used to tie animated graphics to math are capable of even more “aha!” moments.
Yes, I agree on both counts — that I should think about making interactive Java illustrations, and that Jim Blinn’s brilliant animations are not more readily available.
Come to think of it, you might be able to animate the formula directly in a tool like http://www.abettercalculator.com
The Mac’s “Graphing Calculator” from the ’90s (later found at pacifict.com) and this web site are sort of spiritual descendents of Blinn’s work.