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

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.)

from Hacker News https://ift.tt/3x88aun