Finding Musical Beauty in Eulers Identity

Devlin's Angle

April 2007

Finding Musical Beauty in Euler's Identity

This month's column presented me with a major dilemma. Since this month marks the 300th anniversary of the birth of Leonhard Euler, how could I not write about that? Besides being the most prolific mathematician of all time in terms of written output, to my mind, and that of others, this great Swiss mathematician is one of the four greatest mathematicians of all time (the others being Archimedes, Newton, and Gauss). The trouble was, I knew that every other math writer would cover the same story!

So I hummed and ahhed, and hestitated. As a result, the middle of the month came along and still I had not written my column. But the upside of such procrastination is that eventually I noticed that no one else seemed to have written about singing Euler. That's right, putting Euler's mathematics to music. So, finding myself with an opening, here is my contribution to the celebration of Euler's birth.

I have written before in this column ( that I believe Euler's identity

e^(i pi) = - 1
to be the most beautiful mathematical equation of all time, the mathematical equivalent of Da Vinci's Mona Lisa or Michaelangelo's David. Here, shortened and paraphrased, is what I wrote then:

The number 1, that most concrete of numbers, is the beginning of counting, the basis of all commerce, engineering, science, and music. As 1 is to counting, pi is to geometry, the measure of that most perfectly symmetrical of shapes, the circle - though like an eager young debutante, pi has a habit of showing up in the most unexpected of places. As for e, to lift her veil you need to plunge into the depths of calculus - humankind's most successful attempt to grapple with the infinite. And i, that most mysterious square root of -1, surely nothing in mathematics could seem further removed from the familiar world around us.

Four 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.

The beauty I was referring to in that passage is, of course, mathematical beauty. But true beauty challenges us to find different representations, different interpretations. My comparison to music was intended as just that - a comparison. But can that be taken further? Is it possible to really capture (or at least reflect) some of the mathematical beauty of Euler's famous identity in song? I believe the answer is yes, and that it has already been done. Some months ago, a remarkable group of women in Santa Cruz, California, who perform together under the name of ZAMBRA, put their collective talents together to interpret Euler's identity in song. You can hear the result for yourself at where you will also hear them describe the process that led them to their interpretation.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published last year by Thunder's Mouth Press.