What are its uses in the real world?
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Short Answer: the golden ratio is the “most” irrational irrational number
Many numbers in math are irrational, meaning there’s no way to represent them as a ratio of 2 integers. However, often times you can approximate them, such as approximating pi as 22/7. It turns out there’s a way to generate these approximations that gets better and better the more iterations you do. These iterations are called continued fractions and each iteration gets more accurate by some amount, some gain a lot of accuracy, some gain the (mathematical) bare minimum.
So the question people had is, “if we find a number that during this process, gains the bare minimum of accuracy per iteration always, it can be considered the “most” irrational number.” And that number is the golden ratio.
Regarding real world uses, it shows up in biology when a plant choses where to grow branches/seeds to try to make sure they never line up. It also shows up in financial modeling and many other non ELI5 things as well.
Video: https://www.youtube.com/watch?v=sj8Sg8qnjOg
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