Fibonacci sequence

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Can someone please explain the concept and theory behind this?? I’ve heard alot of nature and bees all add up to the ratio but im so confused as to why and whats the purpose.

In: 9

The Fibonacci Sequence is a sequence of numbers where the current number is the sum of the previous two numbers;

0 1 1 2 3 5 8 …

0+1 = 1
1+1 = 2
1+2 = 3
2+3 = 5
3+5 = 8 and so on.

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>Fibonacci numbers are related to the golden ratio, which shows up in many places in buildings and in nature. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone. The Fibonacci numbers are also found in the family tree of honeybees.

[https://simple.wikipedia.org/wiki/Fibonacci_number](https://simple.wikipedia.org/wiki/Fibonacci_number)

The Fibbonacci sequence is a recursive formula. It’s defined by itself. For any given index in the sequence, add up the two prior numbers in the sequence, but start with two 1s.

The sequence is this, for the first bunch:

1 1 2 3 5 8 13 21 34 55

Fib of 4 would be the fourth number in the sequence, 3, because it begins with two 1s. But how would you calculate it?

This is the formula:

* Fib(x) = Fib(x-1) + Fib(x-2)
* Fib(1) = 1
* Fib(2) = 1

Note the two extras. You can’t actually find Fib(4) without also finding Fib(3) and Fib(2) because that’s what “recursive” means: you need to use the formula *within itself* to calculate a value. The “recursion” stops once you hit a “base case”. That is, a known value for a given input. We know that if we give it 1 or 2 we get back 1 (because the Fibonacci sequence has simply been defined that way), so the “recursion” will finally stop. Calculating the Fibonacci sequence is kinda like going backwards until you hit those two 1s, and then you go forward again (and a lot of recursive functions are like this).

Fib(4) = Fib(3) + Fib(2)

Ok, what’s Fib(3)?

Fib(3) = Fib(2) + Fib(1)

Thankfully, we already have values for Fib(2) and Fib(1): they’re both 1.

Fib(3) = 1 + 1 = 2

So now let’s plug those back in to Fib(4).

Fib(4) = 2 + 1 = ***3***

If you just start with two 1s, you should be able to go as high as you want. What number would come after 55? 89, because that’s the result of 34 + 55.

Others have explained what the Fibonacci sequence is, and where it shows up in nature. But you may also want to know – how does nature keep producing that? What’s the reason?

The reason is the simplicity of the rule. The next number is equal to the sum of the previous two. In nature, there’s a lot of things where the next thing builds on the previous thing or the previous several things; when the next thing builds on two previous things, it can naturally produce the Fibonacci sequence or a subset or approximation of it.

The Fibonacci sequence is a sequence starting with 0 and 1, in which the next term is always the sum of the previous two. The Golden ratio is usually what you hear about though when referring to nature, so lets define the golden ratio in terms of the Fibonacci sequence. Listing the first few terms it’d be [0 1 1 2 3 5 8 13 21 34]. A ratio is the amount of times one value is contained by another, also known as a division. So lets try going over the sequence dividing the current number by the previous.

0/1 = 0. 1/1 = 1. 2/1 = 2. 3/2 = 1.5. 5/3 = 1.67. 8/5 = 1.6. 13/8 = 1.625.

This will go on forever, as the Fibonacci sequence is of course infinite. But this division will continue to approach a value of about 1.618. This value is called the golden ratio, or Phi. Like Pi, it has an infinite number of non repeating digits.

Now, why does it appear in nature? Well, when Pi appears in nature its pretty clear why. Its something circular, and Pi is defined by the ratio of a circles circumference divided by its diameter. So if you have a circle, you have pi. With Phi its a bit trickier. The most common place we’ll see it is with things that spiral. Why is that? Well, lets look at another way to express the Golden Ratio, the Golden Rectangle and Spiral.

You can discover it yourself, following the instruction below. If you don’t want to, look up an image of Golden Squares/Spiral and you’ll see this effect.Take a piece of grind paper, and near the center draw a square around a single square. Do the same to the one directly right of it. Above them, draw a 2×2. You should now have a 2×3 rectangle. If we then draw a 3×3 square to the left of them, we now have a 5×3 rectangle. One more time above with a 5×5 square, and you’ll have a 5×8. Keep doing this, going counterclockwise the whole way . Notice how the dimensions of these squares form the Fibonacci sequence with there dimensions. Notice how they spiral outwards, getting larger and larger. If you draw a diagonal line across each square in the same counter clockwise motion, you’ll actually be able see that spiral begin to form. If you use a compass it’ll be even more clear. Seeing this, we can now redefine Phi not in terms of purely numbers, but in terms of something physical. The area of the 5×5 square we drew is 25. The rectangle it was placed next to was 5×3, an area of 15. If we divide 25/15, we get 1.666. If we repeat this, taking the area of the next square, and dividing by the total area we drew before, it will approach Phi!

Getting more real world, where could we see this in nature? Essentially anywhere in which things of a small size build up and spiral out like this we’ll be able to find it. Many flowers form spirals with there leaves or branches in this exact sort of way. They grow naturally in a spiral pattern, with future levels getting bigger. Many plants grow in this recursive way, simply because its efficient. The ratio then comes out naturally, just as Pi comes out naturally with circles. We don’t build tires intentionally to have the Pi ratio, but if you build a circular tire it will express Pi naturally.