They seem the same even the real life examples and I am confused
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Golden ratio is a specific number, about 1.618 , it has various interesting mathematical properties which led ancient Greek artists to be obsessed with it and thus started appearing everywhere in art and architecture (earning it the moniker Golden). ( Later edit: reading up a bit, it looks like the fascination over the golden ratio is a Renaissance thing and it spilled over in their admiration of Antiquity, seeing instances of it everywhere. Careful measurements show that classical Greeks didn’t actually bother with it, they were more into rational numbers).
The Fibonacci Sequence is a, well, sequence: an infinite set of numbers, starting with two ones and appending the sum of the last two values: 1, 1, 2, 3, 5, etc. So definitely different concepts.
What links them is the fact that the ratio of two consecutive terms in the sequence approaches the golden ratio.
Yes, it’s true that the golden ratio appears in lots of places in nature, especially related to growth, but not necessarily because of the way the Fibonacci series is presented (or not directly, it’s fairly simple math and everything is connected somehow). My favorite explanation is how it’s “the most irrational number”: [https://www.youtube.com/watch?v=sj8Sg8qnjOg](https://www.youtube.com/watch?v=sj8Sg8qnjOg) .
One is derived from the other. The golden ratio is what you approximately get when you divide two adjacent, arbitrarily large members of the Fibonacci sequence, one by another. The larger are the numbers, the closer you get to the actual ratio.
Great answers below, I’ll just contribute a useful fact – converting from miles to kilometers follows the Fibonacci Sequence pretty dang well. 3 miles = 5 kilometers, 8 kilometers = 5 miles, etc. Useful for back of the envelope math when you’re traveling around or planning endurance races.
In addition to the other answers:
One of the really beautiful places to see the Golden Ratio in a really awesome form are continued fractions. Here, the Golden Ratio is just the infinite sequence of only 1’s.
Thank you for all the amazing answers!
The fibonnaci sequence is a mathematical formula, defined as such:
f(1) = 1
f(2) = 1
f(x>2) = f(x-1) + f(x-2)
These will lead to the sequence: 1, 1, 2, 3, 5, 8, 13, etc.
The fibonnaci sequence can also be thought of as a difference equations where the new value is derived from the previous states. This is basically a fancy way of saying the the growth of the series is dependent on the previous states. Because of this, we can use the fibonnaci sequence to approximate the Golden Ratio, which is used to describe a lot of natural growth. For an arbitrarily high value of x, f(x)/f(x-1) will produce a value close to the Golden Ratio. For example 13/8 is 1.625, which is fairly close to 1.618….
It has not been mentioned yet, somehow: F_n = [ϕ^n / sqrt(5)], where [] is rounding to the nearest integer.
This is an easy consequence of the (better) formula by Binet, stating that exactly (no rounding!)
F_n = (ϕ^n – (-ϕ)^(-n)) / sqrt(5).
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