Not a lot of ELI5 answers, but some good history.

The Fibonacci sequence is a set of numbers with a distinct pattern (explained in other comments). What is important is that the ratio of one number to the one following it is always the same. (The second is always 1.618 times larger than the previous). That is called the golden ratio, and it is the golden ratio that is seen everywhere in nature.

If you’ve seen the image of rectangles that form into a spiral, this is what it means:

The small rectangle has sides with that exact ratio. The long side of that rectangle is the short side of the next, and that rectangle uses the golden ratio. The long side of that one is the short side of the next…. And so on. This creates a spiral pattern, and that pattern, in that ratio, happens all the time. Flowers, tree leaves, and animal shells for example. Always 1.618 times bigger than the previous part.

The number isn’t magical. 1.618 isn’t special. There is just a natural order to things, and we created a numerical system that happens to measure that order at that number.

In the Fibonacci sequence, each number is the sum of the two previous ones. It is helpful in computer science, for instance, for creating random numbers and sorting data. Natural examples include the spiral shapes of shells and galaxies.

There are multiple ways to define Fibonacci numbers:

• Set the first two to be 0 and 1, and every after as the sum of those two preceding it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... .
• The number of different ways to form a strip of fixed length by glueing strips of lengths 1 and 2 together.
• The number of binary (only 0 and 1 allowed) sequences with a fixed number of digits, and 1s must not be consecutive.
• Via Binet's formula as ( φ^n - (-1/φ)^n ) / sqrt(5).
• [many more]

how it it's supposed to be in all nature and that's sacres geometry...

That's a myth at best, and a lie at worst. There are some very few instances where they somewhat appear, but those are one in a million things. None of the claims of golden ratios appearing within humans, plants or animals has ever withstood scrutiny, sqrt(2), 1.5 and sqrt(3) are just as probable and nonsensical.

Edit: spelling.

φ ~ 1.618 is the golden ratio, satisfying φ^2 = φ + 1. By solving this equation, this means that φ = (1 + sqrt(5)) / 2, where sqrt(5) is the square root of 5.

The formula describes how to get the Fibonacci numbers from φ alone. Actually, it can be simplified a bit: multiply φ a bunch of times with itself, divide the result by sqrt(5), and then round to the nearest integer; you will get a Fibonacci number. Or as a formula: round( φ^^n / sqrt(5) ).

That second term (-1/φ)^n / sqrt(5) is very small, especially if n is large, and is just the "correction" to get to the nearest integer.

Fibonacci was a mathematician who published a book. The entire purpose of the book was to show how much easier it is to do mathematics using Arabic numerals, as opposed to Roman numerals. One example he gave was a simple list of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... et cetera. The sequence is formed by adding the two most recent numbers to get the next number.

Everyone knows that Fibonacci isn't real, it's all a hoax like the round earth or pirates

You can prove that it produces natural numbers through induction for all cases! It is a very fascinating formula indeed, just shows how interconnected all of math is.

the square roots cancel out

It has something to do with the golden ratio

Basically, an = an-1 + an-2 will lead you to solving r² = r + 1, to which the golden ratio is one of the two solutions (-1/phi being the other one and is the number in the second parenthesis)

The Fibonacci sequence is useful for approximately converting miles to km (e.g. 5 mike's ~8km, 8 miles ~13km etc.)

There's also a betting system based on Fibonacci numbers.

This isn't really an application, but the golden ratio also appears in other random places. For example, 1+1/(1+1/(1+1/(1+1/(...)))) is equal to the golden ratio.

There are data structures based on Fibonacci numbers that allow certain actions to be performed more efficiently.

If you keep drawing uniformly distributed random numbers in the interval (0,1), the expected number of draws until their sum exceeds 1 is e.

If you have, say £100 in a bank paying r% interest, if the interest is paid once at the end of the year, you'll have £100*(1+r) in the account. If the interest is instead paid twice a year at r/2% each time, you'll have £100(1+r/2)^2 . However, if the interest is applied continuously instead such that the annual rate is r%, you'll have £100e^r .

This isn't really related to your question at all, but Benford's law is really cool. It says that the leading digits of real-worod data aren't uniformly distributed. The leading digit is much more likely to be a 1 than a 9. This is used sometimes to check whether data has been forged.

It is useful in programming interviews to see if :

a) interviewee can code a recursive routine

b) then recognize that the runtime for a recursive fibonnaci is actually very horrible and to re-write either using loops or memoization.

I'm not sure if this qualifies as "application", but it can be shown that successive Fibonacci numbers are the worst case input for the Euclidean algorithm. So they are useful in the sense that they allow us to analyze the computational complexity of a practical algorithm.

Counting rabbits.

I think it is moreso that it appears in weird places which makes it interesting

Isn't that a continuation of the fibanocci ?

Going backwards.. 5,3,2,1,1,0,1,-1,2,-3,5..

Spiral mechanics are just repeating the same shit over and over. There is no growth, just repetition. It's a trap.

Why do I always get your posts on my feed lol. Interesting though.

Do tell more

(1, 0,) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229

I know this cosmogensis ;)

S(n)=f(v n+1)

Lateralus

Educational trash. Love it

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Mind blown

Doesnt this impy that 1/89 has a non repeating decimal expansion? I thought all rational numbers have a repeating decimal sequence

Gist (might be off by one etc):

Generating function for fibonacci

\sum F_n x^n = 1/(1-x-x^2)

Put x = 1/10

It's because what happens when you put the base into the characteristic polynomial of the Fibonacci sequence

ie

Evaluate x^2 - x - 1 for for x = 10

It also means you can work out both what happens in other bases and what happens for other recurrence relations (eg when the next term is the sum of the previous three terms in the sequence instead of the previous two; that would work out to be 1/889).

Let x be the number above: essentially the sum of 10^(-n) times the nth value of the Fibonacci sequence starting 0,1, ….

The recurrence relation tells us that 10x+x is the sum of 10^(-n) times the Fibonacci sequence starting 1, 2, … which is just the original number shifted left two digits and taking mod 1. so 10x+x+1=100x.

Solve this for x=1/89.

For reference

https://oeis.org/A021093

I would think it shows up in any base b for 1/(b^2 - b - 1)

x + y = xy

x/y + 1 = x

y + 1 = x

y + 1 = x/y * y

y + 1 = y * y

y^2 = y + 1

This is, of course, assuming y ≠ 0

Edit: Oh yeah, golden ratio, not fibonacci sequence

You didn’t get the Fibonacci sequence, you got the golden ratio. Your bottom equation may be written as x = y². Substituting this into the top equation gives

> y² + y = y³

Or equivalently,

> y(y² - y - 1) = 0

Solving for y gives y = 0, φ, or -1/φ, as φ = (1+sqrt(5))/2 and -1/φ = (1-sqrt(5))/2 are the roots of y²-y-1=0. Then using x = y², we get that the intersections are

> (0, 0), (φ², φ), (1/φ², -1/φ)

EDIT: (0,0) is extraneous because y = 0 is not allowed in the second equation.

I think it's because the first equation can be written as x/(x-1)=y and the other as sqrt(x)=y and when you search x for sqrt(x)=x/(x-1) you find phi^2

If you got phi because you want to find it, there’s nothing special. But if you got phi by accident, then it’s a sign. Consult the Oracle of Delphi, pray, or meditate to find out what to do next. Anyways, the math:

Whenever you get a systems of equation, solve for one variable and substitute.

> Solve for x in eq2 gets you: x = y²

> Substitute that into eq1: y² + y = (y²)y

> Move everything to one side: 0 = y³ - y² - y

> Factor out a y: 0 = y (y² - y - 1)

> Solve: y=0 and y² - y - 1 = 0

> Solve the second equation with the quadratic formula : y = (1±√5)/2 ~= 1.618, -0.618

> Substitute the y in an original equation: x = y² = (6+2√5)/4, (6 - 2√5)/4 ~= 2.618, 0.381

So the 2 point of intersection are:

(phi, phi^2 ) and (1/phi, -1/(phi^2))

Also not so sure if this is the right sub to ask if not please redirect me

The difference between the first and second segment is not 1:1, as it always is the the Fibonacci Sequence. It is closer to 1:1.618. Even the boxes in the spirals posted over the photos here do not start with 1, 1, 2, 3, 5, 8..... (or end with ....8, 5, 3, 2, 1, 1). Instead it either starts at a certain size and goes down from there at .618..... from the bigger to the smaller box, or begins with a certain size and increases by 1.618.... from the smaller to the larger one.

The fibonacci sequence is always 1, 1, 2, 3, 5, 8, 13....... So the ratio begins at 1, 2, 1.5, 1.666666 and so on and only approaches the Golden Ratio as it approaches infinity.

The Golden Ratio starts and is always an irrational number like Pi, but it is Phi. Which is 1.61802298875..... and never repeats. Generally it's shortened to 1.618. So, no matter where it begins the next number is 1.61803398875....... more than the previous one.

Things in Nature and most company logos are based on the Golden Ratio and not the Fibonacci Sequence.

Golden Ratio and Fibonacci Sequence

Edit: Phi (sign for Golden Ratio) = a/b = b/c = c/d and so on. Fibonacci does not follow this.

They are related like a dog to a wolf, but you can't call a dog a wolf or a wolf a dog.

The hermetic principles include rhythm and polarity along with some others that are a nice set of patterns found throughout nature.

The universe operates in 2s (0,1), 3s (3,6,9) and infinities(imaginary and irrational numbers) great insight :)

From what I understood, the back and forth of the pendulum is the starting point. It's 2-D, but when you shift to the 3rd dimension it becomes a spiral.

Some resources you may like.!

Stalking the Wild Pendulum - Bentov - OG book. He also wrote brief tour of higher consiousness.

Kybalion. ! Youre on to something. And it is discussed.

You're noticing the idea of recursion: the emergence of larger structures through repetition of a basic pattern.

On a societal level there is not just a duality, but a multidimensional framework that drives behavior, the stock market, the weather, and fluid dynamics. This is known as chaos theory, and is when recursive systems demonstrate extreme differences in outcome based on slightly different initial conditions.

For example, the pendulum is a well defined system, but take a double pendulum and a small difference in the initial position leads to two completely different paths. From this applied to sociology we get the butterfly effect, where a small change in the past can lead to a huge change in the future (i.e. the assassination of Archduke Ferdinand causing WWI).

Despite the chaos in these systems though, there are also overarching patterns such as the 80-year economic cycle noted in capitalism. So on one level everything is random, but on another there tends to be a pattern, which is why history is filled with different players and costumes, but it's all fundamentally the same game repeating in an outward spiral fueled by technological and scientific growth.

The Fibonacci sequence emerges in nature because the golden ratio gives the most efficient of seed packing spirals inside sunflowers and even appears in bee genealogy. While nature itself grows in a way affected by microscopic changes and is hence chaotic, the way leaves branch from a central stem following a spiral or the fractal dimension of trees is consistent.

Where the great complexity of nature arises is in the interplay between order and chaos: the boundary becomes the whole. Just as the coastline between ocean and land is complex and ever changing, life is a boundary condition between order and chaos, that itself is more expressive and self referential than the rigidity of modern computers but more coherent than just the random motion of interstellar dust or star plasma.

In theory, it would be possible to influence future events if you knew exactly how your actions would influence everything in the future, but in practice this is often impossible because the emergent patterns and cycles within these chaos systems are self reinforcing and often are resistant to the butterfly effect.

Chaos theory and synchronity go together. Coincidence is, I believe, an illusion, and can be altered using forces not fully definable in science. Geometry and consciousness taps into a universe force of synchronicity and makes certain outcomes more likely, which is where sigils and sacred geometry derive their power. Consciousness is the inverse of entropy, and structures energy where entropy dissipates energy.