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.

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.

It's because the sequence is usually found in optimized structures and systems. Nature tends to reach it through trial and error, but as long as the more optimal arrangements keep reproducing or surviving more than the least optimal, they will gradually select systems that are closer and closer to one represented by the Fibonacci sequence. As to why the sequence is found in optimized systems, it's because the sequence is an offset of the golden ratio, and that is an important relationship ratio in a lot of different natural systems.

Check out this video and the two after it for a really great explanation about Fibonacci and Fibonacci-like spirals in nature.

Vi Hart is a lovely nerd

It kind of isn't. There are lots of logarithmic spirals in nature, but not all logarithmic spirals are Fibonacci spirals.

EDIT: this isn't to say that Fibonacci numbers never turn up in nature. They are quite common in botany, for the patterns of seed growth in sunflower heads and pine cones, but it turns out that this is because logarithmic spirals with a rate of turn close to the golden ratio pack very efficiently. Also, the relationship between the Fibonacci numbers and the Golden Ratio is slightly deceptive. While the ratio between consecutive terms of the Fibonacci sequence does tend towards phi, this is not a particular property of that exact sequence, but is also true of all recursive sequences of the form

a, b, (a+b), ((a+b)+b)...

It doesn’t. Not really.

It does appear roughly to systems that tend to compound as they grow but even then it’s not exactly the golden ratio.

People tend to misrepresent any logarithmic spiral as the golden spiral.

You can find the number 1.6 everywhere If you look enough.

Tldr: the movie nymphonaniac made us think that any logarithmic spiral is connected to the golden ratio. It’s not. It’s a myth.

There's a joke about spelling and the Catholic church in this somewhere

It’s super common because it’s what happens when you have exponential growth with discrete things.

Exponential growth is when the amount you gain depends on the amount you have. Just like multiplying cells…the more cells that are reproducing, the faster you grow.

But you can’t have half a cell…the integer version of exponential growth, which is basically all cell division or any kind of replication of discrete units, is the Fibonacci sequence. If it shows up in a spiral structure, which is super common for anything that grows attached to its predecessor like shells or flowers, you get a Fibonacci spiral.

I found this guys work a while back http://metatron216.co.za/

You may enjoy it. Also check out the film Pi from 1998.

When taking the number in the ones column only, the Fibonacci sequence repeats after 60.

When you inscribe these numbers around a circle. The cardinal directions are all 0.

Did you see my comment on the holo fractal post on digital addition lol. Looks cool

Very interesting. Is there anything to takeaway from this in particular?

Looks like a game of minesweeper.

Looks like some artwork from Allyson Grey , wife of Alex Grey

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To answer your last question in bold, math doesn't directly prove anything at all about reality. The fact that I can start with some numbers, make up some rules about manipulating those numbers, and then use those rules to generate interesting patterns is cool all by itself. It doesn't have to model anything in reality and often times mathematicians study collections of rules and their consequences without worrying about whether there are any physical processes which behave that way.

Math is just a tool, and sometimes we find things in the universe which behave like some mathmatical structure which we've studied. When that happens, we can use our mathematical knowledge to make predictions about the corresponding processes in reality which appear to be modeled by our mathematical structure. But doing this doesn't prove anything about reality. The second we discover that our model doesn't accurately reflect reality, we need to either abandon our model or enhance it to include new information. But even when that happens, the things we were able to prove about the old model aren't suddenly wrong or invalid. It's just that that particular model no longer applies to the situation. The same model might still apply to some other situation however.

So the answers to your questions are:

1. We can't. And from a mathematical perspective, we don't really care.
2. This isn't really a math question. The word "perfect" has no mathematical meaning at all outside of the notion of a "perfect number". But even in that case, we're just using the word "perfect" to mean something very specific -- that the sum of the proper divisors of the number equals the number. So 6 is "perfect" because 1+2+3=6. But when mathematicians use words like "perfect" in this context, it isn't meant to have any external meaning at all. We could have just as easily used a different word.

2b) I do believe that the golden ratio does show up in art quite a bit, but that's more a question for an art history expert. And as to why it's considered an aethetically pleasing ratio, I have no idea. Certainly there's no mathematical way to approach the question. It's more of a question about human psychology.

3. This is a philosophical question, and not a mathematical one. It's a rather famous and significant philosophical question -- whether math is discovered or invented -- but nonetheless there isn't any way to use math itself to address the question because the words "invented" and "discovered" have no mathematical meaning.

And regarding your contextual paragraph. Mathematicians are often even more aware of how much we don't know than scientists and engineers. Literally every time we answer a question, we end up with 10 new ones. There is no end to the infinite complexity of mathematics and that's just something we have to live with.

How can we be SURE that these numbers appearing in nature aren’t coincidental? 

Sometimes they are coincidence. The closely-related "Lucas numbers" often appear in plants and can be mistaken for Fibonacci numbers. But a lot of the time it really is the Fibonacci numbers and the golden ratio.

I highly recommend ViHart's triple of videos

• "Doodling in Math: Spirals, Fibonacci, and Being a Plant": Part 1, Part 2, Part 3.

In particular, part 2 gives an intuitive idea about why Fibonacci spirals are prevalent in flowers, pinecones, etc., and part 3 adds a biological process to that idea.

is it really that prevalent in art/architecture/ [...] how can we definitively call it “perfect” and “the most aesthetically pleasing ratio” to (at least most) humans?

This is, imo, the most overblown claim about the golden ratio. It's nice enough, but the ratios √2 ≈ 1.414 and 3/2 = 1.5 and φ ≈ 1.618 all have their uses. I don't think the golden ratio looks better than other rectangles.

On the other hand, the golden ratio is also sometimes referred to as "the most irrational number", and there is in fact a precise and rigorous sense in which this is 100% true. If you want to approximate, for example, π = 3.1415926... by a fraction, you could use 31/10 = 3.1 or 314/100 = 3.14 or 314159/100000 = 3.14159, and in general bigger denominators give you better approximations (bigger denominators aren't always better: 22/7 = 3.1̄4̄2̄8̄5̄7̄ is actually closer to π than 314/100 is despite 7 being way smaller than 100). In turns out that if you look at various numbers and how many digits you can get correct from fractions with bigger and bigger denominators, φ is absolutely terrible (it's the "most irrational") in the sense that you need particularly large denominators in order to get remotely good approximations.

Is mathematics a man-made concept, or one we merely discovered?

This is a complex philosophical question that been debated for centuries and has no clear answer. The numbers 2, 3, 5, 8, 13, 21, 34, and 55 really do appear in nature a lot. Formula like "Fₙ = Fₙ₋₁ + Fₙ₋₂" or "φ = (1+√5)/2" are certainly man-made.

are mathematic concepts like the Fibonacci Sequence sufficient evidence in proving anything about the universe?

Different people may have different answers. I would say, "No, and it's not supposed to". Very often mathematics is a useful tool for natural scientists to use, but fundamentally mathematics proves facts about mathematics (not necessarily about the physical universe). Whether the Fibonacci numbers exist in nature doesn't actually affect any of the mathematical results about them.

The reason is because its irrationality lends itself nicely to certain optimisations, which have an advantage over other arrangements. For example, if a plant is growing leaves around its stem and we want to maximise sun exposure, it would be better for the angle between each leaf to be irrational, since that means that they are less likely to overlap (no matter how many leaves, they will still be slightly offset). It is properties like this that make the golden ratio appear in nature.

I may be wrong about this, but I would imagine that we perceive it to be beautiful because it is found in nature. That being said, it being described as perfect doesn't really have any relation to any property of the ratio.

This is more of a philosophical question, so I'll address your later point about reality. Mathematics isn't really a method of proving things about reality on its own. It's only when it is used in the frameworks of other disciplines, like physics, that it really has that power.

What maths does is allow us to start with a set of assumptions (axioms) and derive consequences from this. We can change those axioms, and we can get different consequences.

If you've read this far, you've earned a fun fact that I found fascinating when I first learned it. As you know, the Fibonacci sequence is formed by starting with the numbers 0, 1 (or 1, 1), and adding the previous two numbers together. The ratio between the terms eventually approaches the golden ratio. However, this isn't an inherent property of the starting numbers, but from how you build future terms. In fact, no matter what two numbers you start with (you could start with -11 and 32, or 10000 and 77), and the ratio would still approach the golden ratio.

The "golden ratio" stuff is a hotbed of quackery, due to the kind of pareidolia you're alluding to. It's not everywhere in nature, and its appearance in art is usually deliberate. It sounds exciting if you say that it is, though.

For 3, this is referred to as the formalism versus Platonism debate.

The "discovered vs. man-made" argument is so trite. If you wrote a 10 page paper on that you'd bore your professors to tears.

Mind blown

Educational trash. Love it

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Lateralus

Since it’s the same recurrence relation as the Fibonacci sequence, we have the same general solution f(n)=Cφ^(n)+Dψ^(n). Using initial conditions f(1)=-φ and f(2)=1, we get C=0 and D=1+φ, giving an explicit solution of f(n)=(1+φ)ψ^(n), which sure enough gives f(1)=-φ and f(2)=1. Since |ψ|<1, then powers of ψ get closer and closer to 0, and since ψ<0, the sequence is alternating. Hussah!

can i still put it on r/unexpectedfactorial ?

It's not the Fibonacci sequence in that case. You can still call it a Lucas sequence though.

Omg he really a BALLER

Angle

Now tell lil o to write a program to print fibonacci series in C

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).

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

For reference

https://oeis.org/A021093

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.

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