Is Euler’s Identity Beautiful? And If So, How?

By Keith Devlin @profkeithdevlin

Beauty, they say, is in the eye of the beholder. That’s certainly the case when it comes to beauty in mathematics. I long ago learned that not every mathematician agrees with me when I describe a particular mathematical concept or result as beautiful; they may see it as anything but. 

To be sure, a great many non-mathematicians say they see nothing beautiful in any mathematics, but that’s usually because their only experience of the subject is a miserable school math class experience, which in many cases provides as much exposure to real mathematics as a class on mixing concrete would elicit student appreciation of fine architecture.

But even among those who know, and are proficient in, advanced mathematics, people can differ in what they find beautiful, just as we find in the art world. I was reminded of this recently when the tweet shown here landed in my feed. The author, Francis Jeffries was known to me; in particular, I knew him to be an accomplished scientist who, among other things, has worked with creative artists (most notably, Arthur C. Clarke).

You can see my immediate response below Jeffries’ tweet in the brief thread, but Twitter is far too simplistic to get into anything as deep and profound as mathematics or beauty. All I did there was raise the issue of perspective: Beauty is in the eye of the beholder. Let’s take a somewhat deeper dive into that.

The most beautiful equation in the world?

The focus of Jeffries’ tweet was the Euler Identity, shown here.

I wrote about the beauty I see in this identity in an article published in the Wabash Magazine in 2002, in conjunction with a guest lecture I gave at Wabash College’s Center for Inquiry in the Liberal Arts.

Having spent twelve years in the middle of my career at elite liberal arts colleges (Colby College in Maine from 1989 to 1994, then Saint Mary’s College of California from 1994 to 2001), I pitched my talk at the student audience I would be addressing at Wabash, and aimed my article at Wabash College alumni.

Brief aside: One decision I always make when speaking to, or writing for, a mathematically lay audience about this topic is to use the term “Euler’s equation,” rather than the more accurate “Euler’s identity” that professional mathematicians use. The expression involves numbers and an addition sign, and has an equals sign; to most people, that’s an “equation.” Why risk causing confusion using a different term? (For the record, an equation has one or more unknowns, that you have to solve for; an identity involves only constants and expresses a definite relation between them. An equation challenges you to solve it, whereas an identity states a fact that may surprise or impress you. (Though it can be quite a challenge to understand why an identity is valid. As is the case with Euler’s Identity.)

One passage from my Wabash article has been circulated widely ever since it appeared:

“Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”

The preceding passage, which provides the setup for that “liberal arts college targeted” quote, indicates where I was coming from in describing the Euler Equation as beautiful. Referring to the mathematical constants 0, 1, pi, e, and i, I wrote:

“Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.”

In writing my article, I knew from experience that members of a mathematically lay audience in a mathematics talk can be assumed to be familiar with 0 and 1, the identity elements for integer addition and multiplication, respectively, and with pi from their geometry or trigonometry math class; if they got to calculus, they will have encountered e; and somewhere along the way they will have heard about (but most likely not looked at in any depth) that mysterious “imaginary” number i, the square root of  –1. But they were unlikely to know much beyond that. That made them wide open to the pitch I was going to make, and the way I would frame it.

Given the fundamental nature and wide applicability of those five constants in mathematics and many of its applications in physics, engineering, economics and finance, and elsewhere – and noting (1) that there are no other mathematical constants a student typically encounters in their school education and (2) their origin in very different parts of mathematics, there is a significant surprise value in discovering that they are all bound together is a single, very simple equation (identity). Directly following that much-quoted Wabash passage, I elaborated:

“It brings together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate.”

This may have been a pitch directed at a particular audience, but it was nonetheless genuine. In writing my article, I was drawing very much on my own experience in high school, when I first came across the Euler identity. It quite simply blew me away. 

Looking back later in my mathematical career, with an awful lot more mathematical knowledge under my belt, I realized that my earlier surprise and joy was a result of my limited understanding at the time of the five numbers the identity links. By the time I had graduated with a bachelors degree in mathematics, I could see it was all smoke and mirrors; a spectacle that can work only on an audience that does not have sufficient knowledge of the system of complex numbers, developed in the 19th Century.

To the more knowledgeable, post-Ph.D. me, Euler’s Identity is, on the face of it, just a big yawn. But appearances can be deceiving. The significant issue, and where we find the far deeper beauty, lies in why and how that earlier mind-blowing mystery had become a nothingburger. This is “Mind of God” stuff. And the mystery only deepens.

Before I go into that, I should note that I repeated some of my Wabash article in a post about Google I published in Devlin’s Angle back in October 2004, just after they became a public-traded company. There, I wrote (also with lay readers in mind):

“To me, this equation is the mathematical analogue of Leonardo Da Vinci’s Mona Lisa painting or Michaelangelo’s statue of David. It shows that at the supreme level of abstraction where mathematical ideas may be found, seemingly different concepts sometimes turn out to have surprisingly intimate connections.”

The second sentence of that observation brings us to what I – the mature professional mathematician of today, not the eager young math student in my final year at high school – see as the “real” beauty of the Euler identity. The beauty that is not a result of smoke and mirrors. (Or is it? That question will lurk as a subtext about mathematics to all that comes later in this essay.)

The celebrated British logician and philosopher Bertrand Russell (1872-1970).

And this is the moment where I bring in that famous Bertrand Russell quote! (I know you were expecting it. I will not disappoint.)

Writing in his book A History of Western Philosophy, long before text-printed mugs and t-shirts were invented, he opined:

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.”

Let’s try to see where he was coming from, and what he was attempting to convey.

To fully appreciate great art, you have to get beneath the surface features, where the artist has buried the gems. Appreciation, and the delight that comes with it, has to be earned by way of the experience of finding your way into what the artist was trying to convey. 

Actually, that’s not entirely correct. The work of art you see or otherwise experience is the end-point of the artist’s own journey of discovery and understanding.

And that’s the case with the deeper beauty we find in mathematics. Trying to understand the tantalizing mystery Euler’s Equation presented me with, when I encountered it as a 17-year-old high school student, was one of the stimuli that drew me into the world of (advanced) mathematics. It took me an entire three-year education as a mathematics undergraduate before I could look at Euler’s identity and say, “Yeah, it’s no big deal. I just did not know enough five years ago. I see it all now.”

The mystery had gone, replaced by deeper understanding. And there, with that very personal, emotionally rich, intellectual journey of discovery that we all must follow if we wish to understand mathematics, we find the deeper beauty of mathematics, the beauty Russell was surely trying to articulate when he wrote that mathematics “possesses not only truth, but supreme beauty—a beauty cold and austere … sublimely pure, and capable of a stern perfection.” 

Yes, Euler’s identity is beautiful, for all the reasons I described in my Wabash article. That beauty can be appreciated by anyone with a high school knowledge of mathematics. And for someone whose mathematics education got no further than that, the identity’s beauty also comes with a mystery: Why is it true? What is going on? So it’s a mysterious beauty. What could be more alluring?

To someone with a bachelors degree in mathematics on their resume, however, the mystery gives way to a far deeper understanding. That person now knows why. And there too is beauty, but a beauty of a different kind, the beauty of the hitherto hidden mathematical structure that connects those five constants – the “supreme beauty,” to use Russell’s term. But eliminating the mystery does not destroy the beauty originally seen; how could it? Rather, that original, “surface” beauty is enhanced by the appreciation of a far deeper, structural beauty.

So too, when we look at someone we love, we see beauty. Some of that beauty may be a result of resonance with physical form and shape – bilateral symmetry is known to play a role; but to discern their true beauty, we gaze into their eyes to see the person inside that physical form.

At its heart then, mathematical beauty is not really any different from other kinds of beauty. Beauty is always very much in the eye of the beholder.

POSTSCRIPT: Three months before I wrote the Devlin’s Angle post I referred to above, in April 2004, I posted another essay on Euler’s Identity, where I reported on the attempt of a choral group, Zambra, to find a musical interpretation of the famous identity.

Inspired by their work, soon after my essay appeared, I joined forces with them to develop explorations in song of a number of well known mathematical identities, seeing how we could capture their “cold and austere” Russellian beauty with the musical purity of the human voice. After a number of stage performances together, we recorded the show we put together, Harmonius Equations, so others could make the journey with us. You can find the whole thing by following the link. Enjoy! And soak in the beauty …