The Most Beautiful Equation

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"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."

-- Keith Devlin

What a psalm to the great Euler’s Equation. Some people look at the above equation superficially and find it boring, but if only you knew how universally applicable this equation is, it is anything but. Hopefully you will come to the “aha” moment after you finish reading this post.

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The Dream Team of Numbers -- $e^{i\pi}+1=0. $

The trigger for this blog post was actually from the TED talk I put at the end of the previous post. In the video, Jim Simons mentioned Euler’s equation. He pronounced it in a way that sounds unnatural but familiar to me (déjà vu I thought).  Then I searched the term in google for pronunciation and realised I actually have heard of the name ‘Euler’ in Chinese. Since I remember Euler about nothing but the name, probably from high school maths class, I decided to refresh my memory a little bit. Many articles on the internet turned out to be engrossing readings!

It would be unfair to introduce Euler’s equation without first introducing Euler the great mathematician. Leonhard Euler was an extremely prolific 18th-century Swiss-born mathematician, who has 886 papers and books published under his name, according to the U.S. Naval Academy. He did so much work that many of those were published posthumously for more than 30 years. According to the U.S. Naval Academy, just in mechanics there are “Euler angles (to specify the orientation of a rigid body), Euler’s theorem (that every rotation has an axis), Euler’s equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations).” One of his most important contributions that we will talk about today is the Euler’s Identity. It is simple to look at yet incredibly profound. It is composed of the five most fundamental components in mathematics:

  • $latex e $: The irrational number 2.71818... It is the base of natural logarithms that governs the rate of exponential growth. It pervades in maths in a vast number of equations. (One fun fact: it equals to the sum of the inverse of all factorials from zero to infinity!)
  • $latex i $: It is defined as the square root of negative one: $latex \sqrt{-1} $, the root for all imaginary numbers. It is the "imaginary number", because how on earth can a number multiplied by itself and still remain negative. However, it is very much needed as a stand-in to many situations as a maths trick.
  • $latex \pi $: The irrational number with unending digits 3.1415926..., which is equal to the circumference of a circle divided by its diameter. (More fun facts? Go check: What Makes Pi So Special? )
  • 0 & 1: These two numbers are not only fundamental to mathematics, but also to philosophy and computer science. All human knowledge is built upon the basis of this two-term discrimination. Computer memory uses zeros and ones to store all the information.

Euler’s equation stems from the interactions of complex numbers, and has such wide applications that it can mean many different things in different scientific fields. In this post, we illustrate the basic idea of it by using an example of its application in computer graphics. I’ll let you appreciate my masterpiece for 30 seconds first…

File Apr 11, 14 00 09

OK, now don’t be too carried away by the two little cute lions. The upper right lion was actually cloned from the bottom left lion, except that it has been “Eulerred” (dilated and rotated).  For each lion, I drew a coordinate around them. The horizontal axis is for real numbers, and the vertical axis is for imaginary numbers. The coordinates for the upper right one is obtained by multiplying the complex number $latex 4+3i $ to the coordinates of the bottom left one (check that using the coordinates I wrote for its feet). The effect is that the lion has been enlarged 5 times, and rotated anti-clockwise by 36.9 degrees. The reason it rotates 36.9 degrees is self-explanatory in the following plot. In the complex number $latex 4+3i $, 4 specifies the direction of the move horizontal-wise, and $latex 3i $ vertical-wise. Together, they specify the degree of the rotation, and the size of the dilation. $latex 4+3i $ is called the rectangular form of the complex number, and we can also transform it to the corresponding polar form $latex r e^{i\phi}=5e^{i0.644} $ (0.644 radian is equal to 36.9 degree), which now resembles one part of the Euler’s identity.

File Apr 11, 14 33 32In fact, Euler’s identity is just a special case of $latex a+bi $ where $latex a=-1 $ and $latex b=0 $. So in polar form, we have

$latex r e^{i\phi}=a+bi \Rightarrow 1\cdot e^{i\phi}=-1+0i \Rightarrow e^{i\phi}=-1\Rightarrow e^{i\phi}+1=0 $.

Voila! I know I took for granted the fact that the given rectangular form gives us the corresponding polar form, because the derivation really involve a nontrivial amount of calculus. Starting by playing around with a bit trigonometry:

$latex a+bi=(r \cdot cos\phi+r \cdot sin\phi)i=r(cos\phi+i \cdot sin\phi)=r\cdot cis \phi $,

and further derivations are shown below on the equivalence of $latex cis \phi$ and $latex e^{i\phi}$ (if you bother to look into the nitty-gritty).

[caption id=”attachment_510” align=”alignleft” width=”1700”]equa Credit: Robert J. Coolman[/caption]

There are way more applications of Euler’s equations, such as Euler Differential Equations in fluid dynamics and Euler’s equation in Fourier analysis for signal processing, just to name a few. Again, at the end I highly recommend you to watch the interesting video by William Dunham (he even specialises in Euler!):

I didn’t mean to have one video at the end of my every blog post as a routine, but hopefully you enjoyed it. So much of a tribute for the crown jewel of mathematics. Have a “Good Friday” my friend!


References:

[1] https://en.wikipedia.org/wiki/Euler%27s_identity

[2] http://www.livescience.com/51399-eulers-identity.html

[3] https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/