What does the word “beauty” bring to your mind? A pretty face, a flower, a mountain range, or even a sound can be easily described as beautiful. While I won’t disagree with any of these, today I’d like to share a few things that I think are beautiful.
The ideal gas law
PV=NkT. This simple equation combines several laws (Boyle’s law, Charles’ law, Gay-Lussac’s law, and Avagadro’s law) into a simple, concise law that explains the behavior of ideal gases. An ideal gas is a mixture of particles where each particle takes up zero physical space and none of the particles interact with each other. Now, obviously real gases don’t act like this, but the ideal gas law is a widely applicable equation.
But the ideal gas law isn’t necessarily beautiful to me because of its applications. The most stunning thing to me is that the ideal gas law can be derived in so many different ways. Originally, in the late 1600s, chemists began exploring the things like pressure, volume, and temperature. They derived the ideal gas law piece by piece using careful experimental techniques. Later, in the late 1800s, Boltzmann and Maxwell used a statistical approach to describing gaseous particles. The beauty of the ideal gas law is that both experimentally and statistically we get the same exact answer. We don’t get an answer that’s close to the same, we get the exact answer.
No essay on scientific beauty is complete without a mention of Euler (pronounced “Oiler”). This simple identity contains some of the most important constants in all of mathematics:
e: This constant is equal to ~ 2.71828. It is the base of the natural logarithm. It’s a constant that appears when describing just about anything in the universe. Also, ex is it’s own derivative, which is pretty awesome.
i: An imaginary number, i is actually just the square root of -1. There is no real number that, when squared, will produce a negative number. Therefore the square root of a negative number is an imaginary number. However, this imaginary number is very real, and things like quantum mechanics (a very real thing) wouldn’t be possible without i.
π: We all know pi. 3.14159. Pi is the ratio of a circle’s circumference to its diameter. It’s another mathematical constant that just keeps showing up (even when you’re not even describing a circle, which is pretty cool).
0 and 1: These two numbers are both interesting mathematically. Zeros and ones can either make a derivation simple (by “getting rid of” a bunch of terms) or insanely complex (a zero in a denominator is never something you want to see…). Ones and zeros make up binary, the complex, but elegantly simple way that we are communicating right now. These two numbers are just as beautiful (and important) as any of the other constants in Euler’s identity.
Now, we know that Euler’s identity is valid because it’s actually a specific case of a more general, and equally beautiful equation:
Once again, the beauty of this equation is not only in its simplicity, but the fact that we can derive this expression using several different methods. Just like e and π in Euler’s identity, Euler’s formula is used over and over again to help us derive some of the most important scientific equations we know.
Beauty is all around us, and I’m not just talking about trees, birds, and rivers. The universe is extraordinarily complex, but amazingly simple at the same time.