Humans are Great 8: Feynman’s Mirror

One of the unfortunate things we humans tend to do is rate a genius for invention as superior to a genius for explanation.  We stand with (rightful) awe before the original insights of a Bernhard Riemann but shrug off the efforts of people who took brilliant but convoluted existing ideas and found a way for the mass of humanity to gain some purchase on them.  But if something like calculus, which stumped a continent at its first unveiling, is second nature to sixteen and seventeen year old high schoolers now, it is largely because of those people who had a genius for reforming the clunky and abstract into something graspable but still faithful to the rigor of the original.

To be either a creative or explanatory genius is quite enough to earn our dazzled esteem, but to be both is to enter a slim minority of world figures indeed.  Charles Darwin was one such, and I would rank English mathematician GH Hardy as another, but for most science-y people, if you say the words “brilliant explainer” and “genius scientist” in the same breath, they will respond, “Oh, you mean like Richard Feynman?”

And deservedly so.  Yes, he’s been rather – merchandized – as of late, and with that over-exposure has come something of a backlash.  “Oh, Feynman?  I’m so done with that guy.”  But if we step back, away from the t-shirts and novelty coffee mugs, maybe we can recall for a bit what made us fall in love with him in the first place.



For me, there is no better demonstration of him at his very best than the Mirror Example in his QED series of lectures.  It is the quintessence of everything admirable about Feynman’s mind – the ability to take a vastly thorny concept and craft a physical example that retains all of the essential features of the original while smoothing out the parts that contribute formally but not comprehensibly to the whole.

What Feynman is trying to illustrate with the example is how Quantum Electordynamics weighs and combines different possible interactions for a given set of particles to calculate expected observable values.  He asks us to consider how a mirror works, and starts off the way every good science explanation should, with a confidence builder.

He reminds us of the law of reflection, which says that the angle of reflection for light bouncing off a mirror is equal to the angle of incidence (the angle it came in at).  “I remember that!” we all say, and feel that excited willingness to push on that only comes with an initial burst of confidence.  Also, we now have an anchor to come back to if we feel ourselves getting lost.  These are fundamentals of good explanation practice that Feynman just intuitively felt, and are what make him so compelling to read still, a half century on.

Having established a solid base, he starts branching outward.  What if I told you that, in fact, the path where the angle of incidence equals the angle of reflection is just one possible option, the one that takes the least time to travel, granted, but that there are many more paths which light can, and does take?


We get excited – something that we knew for sure was right turns out to have a little bit of devil living inside of it.  And Feynman uses that excitement to start talking about probability vectors, something that most people wouldn’t have immediately found themselves interested in, but that now, eager to resolve the mystery, they will pay rapt attention to.  He tells us how to construct vectors for different possible pathways (say, A-D-B or A-E-B in the above figure from the foundational Feynman Lectures), and uses those vectors to construct a total picture of all possible reflections off the mirror:



What was an obscure concept involving vector addition and complex exponentials thus transforms, through his flair for turning mathematical machinery into physical representation, into this picture which beautifully represents how reality works.  Yes, there are lots of alternate pathways, but the ones at the edges of the mirror tend to cancel each other out, since they all point different ways, so the behavior that we witness is primarily created by the middle of the mirror, where the angle of incidence equals the angle of reflection.  So, at the end of the day, the law we learned in high school is largely true, from a certain point of view.

But then, like all good magicians, he saves his last trick for the moment when we feel comfortable and reestablished in the world.  We can, he informs us, by scraping away the parts of the mirror that cancel out the contributions from the edges, make those edges contribute again.  So, if we wanted, we could purposefully construct mirrors that break the law of reflection after all.  Thank you, quantum mechanics.

And so, from safety, through excitement, to comprehension to safety to daredevilry, Feynman has taken something outside the veil of everyday thought and brought it home to us all.  It’s that willingness to take some time to work on MERE explanation that I love about him, and about those generations and generations of teachers who sit up at nights trying to find new ways to illustrate our scientific heritage to coming generations.

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