The Galactic Golden Ratio

Hello!

Today we will talk about the golden ratio (spoilers: it's not golden), which can be considered as one of the most important and intriguing mathematical constants in nature.

If you read this blog, you may have seen how a spiral galaxy looks, maybe its a swirly kind of structure that seems to have extending arms; basically a spiral. Now I'm going to tell you about something totally unrelated: What is the positive root of the quadratic polynomial \(x^2-x-1\) (Don't worry, I'll explain the connection)?

Firstly we will deal with finding the positive root of the equation. We can use the quadratic formula as follows:

\(\frac{1+\sqrt{(-1)^2-(4)(-1)(1))}}{2}\)

\(\frac{1+\sqrt{1+4}}{2}\)

\(\frac{1+\sqrt{5}}{2}\)

And that's our answer! Now you might wonder "why did we do this"? Don't worry, I will show my very amazing magic trick now.

If you have studied math (I hope you did), you'll probably know about "Pascal's Triangle" and the concept behind it. We don't need to know much about the triangle, just the pattern. Lets take a number \(x\). Lets suppose \(x=a+b \mid a > b \wedge \left \{ a, b \right \} \in \mathbb{Z}\). Now if I want to generate the next number in this sequence; we'll call it \(y\). Now we can define \(y=x+a\) because \(a > b\). And then we can further define \(z\) as \(z=x+y\) and so on. We can define a ratio between the two numbers such as \(R_{1}=\frac{z}{y}, \frac{y}{x}\) etc. Basically, the magic behind this is that when the numbers tend to infinity (which means they are extremely close to infinity but not infinity), the ratio of the two infinitely large numbers that obey the operations shown equals \(\frac{1+\sqrt{5}}{2}\). That is wonderful! We were able to define a relation between that quadratic equation and the Pascal's Triangle!

Now you might wonder how on earth (or well... the universe) this is related to spirals at all, but hold on. The image shown below can perfectly describe what I intend to say.

Diagram created in desmos.com/geometry.

This shows the spiral diagram joined by connecting opposite sides of Pascal's Triangle obeying squares with arcs.

Yes, you see that correctly, the simple innocent numbers that you saw lead to some of the most intricate patterns in nature and astronomy. Now do you see the golden ratio in the galaxy? Maybe also look down. Hurricane, nautilus shells, sunflowers; all of them have this spiral pattern! It seems to be that the golden ratio (which is defined as the irrational number \(\phi = 1.618...\) but no one cares) is everywhere and now once you know about it, you can't not see it everywhere.

Now you can finally understand why everyone talks about the mystical golden ratio and treat it as a god. Oh, and also about the spiral structure of galaxies, yeah...

Thanks for reading!