Why is the Fibonacci sequence found everywhere?
In: 2220
Tl;dr: It is not found everywhere, it describes what is found everywhere.
The way our Universe works is pretty much „Growth“; Atoms „grow“ into Molecules, which grow into Matter, etc. Seeds grow into Trees, Babies grow into Adults, Tribes grow into People.
The way Growth works is always the same: replication.
So if you start with 2 of anything and start to grow a population you will first create an offspring of each pair and add it to the total population – 2 becomes 3, now with 3 you can create 2 pairs, so you get 2 offsprings and add them to the population – 3 becomes 5, which can make 3 pairs and so on.
At some point mathematicians observed this and quickly figured out a pattern. This pattern became the fibonacci sequence.
So in the end the sequence isn‘t something that we invented and happens to be found everywhere, but the other way around: we observed something that happens everywhere and found a way to describe it.
Hope this helps, and sorry for any error or bad wording, english isn‘t my first language.
Why does Pi show up everyone? Well it describes circles, and circles are everywhere in nature.
Same with the Fibonacci sequence (specifically in form of the golden ratio); it describes the natural growth of spirals, and spirals are all over nature; galaxies, shells, flowers, leaves etc.
For example, leaves of plant will grow at the golden ratio, because it does not repeat uniformally and therefore each leaf will receive the most amount of sun. Evolution favours this, so we see it all over nature.
And since its all over nature, humans have evolved to see this Golden Ratio / Fibonacci sequence as aesthetic, comforting and pleasing to the eye. That’s why it’s used in art, design, photography, architecture etc making it even more “everywhere”.
Long story short with no technicality behind it:
The Fibonacci Sequence is made up of numbers equal to the sum of the two previous numbers in that sequence. It essentially adds itself to itself, and so grows proportional to it’s previous self.
The most famous example of the nautilus she’ll is an example of a spiral-esque shape that slowly wants to expand it’s diameter to accommodate for an ever increasing living being while curling around itself. The current number/size of opening grows slightly larger than it was before over time.
If you look at the entire piece from outside you get a Fibonacci sequence-ish.
It’s actually the golden ratio, which is very close to the ratio of Fibonacci numbers as you get up there.
And the oversimplified way to create it starts with a special rectangle, with a length of about 1.6 and a width of 1. Rotate it a quarter turn and scale it to make the width what the length was (~1.6 times as big). This becomes the new width, and the new length scales up too.
Rinse and repeat for each petal on the flower, each node on a branch, etc.
Again, oversimplified because eliactually5.
Actual mathematician here: the answers so far are all wrong, and so is your presumption. Fibonacci numbers are rare and there is no inherent mechanism in the universe for them.
Multiple answers have claimed it is all over nature, but examples of them appearing at all outside human works are pretty rare. They somewhat happen in sunflowers sometimes, if all the randomness of growth does not almost certainly screw it up. But that’s about it. Beyond those rare few examples, it is usually esoteric and/or made up.
All the spiral patterns are not Fibonacci based, either. They are simply what we call _logarithmic spirals_, which have effectively nothing to do with the numbers. They are based on exponential growth, but that could be the powers of 2, 3, the golden ratio (~ Fibonacci numbers), 5, pi, and most importantly and most common, e.
When you encounter individual Fibonacci numbers, it is random chance. Especially with small numbers like 1,2 and 3, they just as well could be a million other sequences. And it it looks like it might be the golden ratio, it almost always might just as well be 1.5, 1.6, square root of 2 or 3, or a lot of other options. The uncertainty is usually very high, and often we even know that it is definitely _not_ that one number.
Some posts even compared the golden ratio and the Fibonacci numbers to pi. But unlike the former, pi has a lot of reasons to be everywhere in a physical reality. For example, the laws of nature do not change when you rotate things, hence a lot of optimal arrangements are ones that don’t change with rotation as well: circles and spheres. Thus pi.
The Fibbonachi sequence approximates and tends towards the Golden Ratio.
The question then is why the Golden Ratio shows up so often. It doesn’t show up *everywhere* but it does show up with unusual frequency compared with literally any other ratio of exponential growth. Why ~1.61? Why not 2.7 or 5.5 or 1.09?
If something occurs repeatedly when it would otherwise be arbitrary or random that generally means some process must be driving a bunch of different things towards the same target because it is optimal. So what makes the Golden Ratio Unique, and when is that property going to be beneficial?
The Golden Ratio is the **most* irrational of irrational numbers. It is the irrational number least well approximated by any integer ratio. If you use it to drive or sample a periodic process you will get the most aperiodic results.
So it makes sense that the golden ratio will show up in the real world in cases where periodic, repeated structures or events are detrimental to some process.
The classic example is growing branches for leaves on a plant as the stem grows upward. If the biological component that sets a new branch to sprout from the trunk spins around as it grows upwards, how often should it deploy a new branch?
Once per revolution would cause all the branches to be on one side of the tree, on top of eachother and blocking the light to those below. That’s bad. Maybe twice per revolution? Now you just have two stacks of branches on opposite sides of the tree – not much better. Maybe 3x? 5x? Any integer rate per revolution will just give you spokes like a bicycle wheel. What about a fraction like 3/5ths? You’ll still get spokes equal to the denominator. Any rational fraction is periodic and overall not optimal.
So try an irrational ratio! Every pi rotations, place a branch! Now no branch will ever sit directly above another branch. That’s an improvement, but pi is really well approximated by 22/7, so it’ll just look like 22/7 with a very slight fanning out of each spoke.
It will hardly be any better than 22/7. The Golden Ratio is the Least Well Approximated irrational number, so it will be the least periodic and get you as far from this ‘spokes’ problem as you can get.
So it’s not unsurprising that many plants deploy branches and leaves and petals and seeds at rates close to this ratio. This isn’t universal by any means but it’s an optimal arrangment when things benefit from not being periodic and thus many things evolve towards the same ideal.
Check this out: cicadas have a dormancy period where they are in the soil for some number of years and then hatch. Different cicada species have different durations they stay in the ground, but they are all prime numbers. That way the birds can’t evolve to eat cicadas every N years.
Cicadas are better at math than birds. QED. /s
I already made this comment but I thought it might need its separate space.
Basically, the answer you get depends on who you ask. Ask a mathematician and you get the top comment. Ask a philosopher, and you get a reply to the mathematicians comment.
As a statistician myself, all I see is random noise making sense ever so often. Throw a bunch of large numbers into the air and there is a probability that they’ll land exactly describing your geographical coordinates. Different phenomenon just have different probabilities of happening. Doesn’t mean that there’s anything special about them. Yes, we don’t know why some things occur with a higher probability than others, but that still doesn’t grant them any special place, not more so than any other random sequence.
Now the question of significance of these patterns is quite tricky and I believe originates from the time where numerology was all everybody was taking about. Even until now, numerology keeps seeping into number theory every now and then. A somewhat similar example would be like chiropractors vs physio therapists (oops!)
In the end, it all seems from our innate desire to find meaning in all things. Which is why sometimes clouds look like objects and you can swear you saw a face on Mars that was just a rock. Our brains are hardwired to find patterns, and more often than not, we just give meaning to something that’s inherently random noise
It’s not, but what it is, is roughly exponential, and exponential growth (or decay) happens when the rate of change of something is proportional to the amount of that something, and **that** is something that happens in many different settings.
So it’s easy to shoehorn Fibonacci numbers into lots of situations by scaling and rounding.
It doesn’t. Here is a web archive link with more details (original link is dead unfortunately): http://web.archive.org/web/20160309052601/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
Latest Answers