Euler’s identity: a mental segue

Working with code as a creator, configuror (is this really a word?), or troubleshooter has rewards and costs. One of the rewards is seeing / discovering crisp and novel logical patterns used to translate business processes from physical to digital.
Recently I was discussing with somebody how best to present large amounts of somewhat abstract data related to individuals with a goal of ‘perceiving’ subject matter experts in as fluid a fashion as possible. (Clearly I struggled with this sentence.) Musing for a few days on the topic I started to visualize a way to map each persons data points onto a sphere and what the values of those data points could, and should, do to the sphere and its surroundings. Then I remembered something I had not thought or heard anybody mention in a long long time. Euler’s identity.
Euler’s identity is not something most people think or even know about. Considered by many to be exceptionally remarkable for its mathematical beauty. It is often mentioned as the literal pinnacle of mathematical elegance and beauty. And it looks like this:
e^{i pi} + 1 = 0,!
It comprises of three basic arithmetic operations that occur exactly once each:
  • addition
  • multiplication
  • exponentiation.
The identity also links five fundamental mathematical constants:
  • The number 0.
  • The number 1.
  • The number ”π”, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
  • The number ”e”, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
  • The number ”i”, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.

So what does that mean? Euler’s formula demonstrates, albeit for still not terribly clearly known reasons, that there is an intrinsic connection between complex exponential functions and trigonometric transformations.

So… what does that really mean? Sometimes other people put it best. “The true beauty of Euler’s Identity comes from the fact that, while the true nature of several of these constants continues to remain a mystery to mathematicians (though it is clear that they possess many real-world manifestations), within the confines of this equation they all work together in such a way that they interlock like pieces of a mathematical jigsaw puzzle, the end result of which has the mathematical traveler ending up right back where he began – at the journey’s origin.
If it does not ‘make sense’ don’t worry. The mathematician Carl Friedrich Gauss supposedly said that if this formula was not immediately apparent to a student upon being shown it, that student would never become a first-class mathematician. Few people are. I know I’m not. But that still does not help if you cannot “see” it, so here’s a picture. A moving one that hopefully will act to shed some light on what Euler’s identity tells us. (I got the picture and the text below straight from Wikipedia.)

The exponential function ez can be defined as the limit of(1 + z/N)N, as N approaches infinity, and thus e is the limit of (1 + iπ/N)N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)N. It can be seen that as N gets larger (1 + iπ/N)N approaches a limit of −1.

If you’re still not sure what it is, Midnight tutor has a good video here:

In case you were wondering. No, it did not help solve the original business need. But it did help to continue to keep alive the awareness that everyday people I work with produce clever, and sometimes stunning, solutions with logical blocks. It’s like working with artists who craft in ether.