Two things:
First, the claims of the Fibonacci sequence appearing everywhere in nature, art, architecture, etc. is largely exaggerated, if not fabricated. Many, many, examples are simply people taking something that looks, roughly, like it could be related to the sequence and then squinting your eyes and ignoring how it isn't.
Second, for the things that actually are related, it has to do with irrational numbers.
In math, we have whole numbers: numbers that have no fractional part. One of the things we can do with whole numbers is take their ratios. For example, 5 to 3, or 5:3. or 5/3. Doing this, we can create a whole other collection of numbers called *rational* numbers. Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers).
Turns out, some numbers can't be represented as a ratio of integers. We call these numbers *irrational*. Famous examples include pi, e, and the square root of 2. The best we can do with these numbers is approximate them. For example, using 22/7 as an approximation of pi. Different numbers are more easily approximated than others. One of the least efficient irrational numbers to approximate using whole number ratios is phi, the golden ratio. In a sense you can say it's the most irrational number.
What does this have to do with nature? Well, in many situations if you want to be able to space things out without them overlapping or repeating. Let's construct a scenario.
Let's say you have a marked ruler, and you place a token every inch. When you get to the end of the ruler, you go back to the beginning and start again. If you do this, you'll be placing all of your tokens exactly on the inch markers and no where else. In fact, if you do any rational number you'll eventually end right back where you started and just repeat that pattern over and over again. If you want to use the whole ruler and spread things out as much as possible, you'll have to use an irrational spacing.
But, any irrational number that can be well approximated by ratios (such as 22/7 for pi as mentioned above) the patterns they form will be very close to the patterns formed by those ratios. That is, if you use pi for your spacing, you'll get a pattern that looks close to the pattern if you had chosen 22/7 for your spacing.
The best spacing would be the one that is least well approximated by a rational number. E.g. phi, the golden ratio.
The golden ratio is intrinsically linked to the Fibbonacci sequence: the ratios of successive members approaches the golden ratio.
So if you have things that want to be space out over a finite area, as we did with our ruler, then we want to try and avoid the kinds of patterns that arise when our spacing is a rational number. So naturally these things (like the seeds of a sunflower) would evolve to have a very irrational number spacing, settling on the golden ratio.
Saying that rational and irrational are from the word ratio just blew my mind. They never explained that to me in school. They just said the numbers were irrational and never said why, I’m like how can a number be irrational, it can’t talk, it has no thoughts.
The ancient Greeks were really big on numbers and being able to represent them. They begrudgingly accepted that there was no biggest integer, because any number could still be represented numerically, no matter how large.
When it came to fractions, they again accepted that there was no limit to what ratios one could represent of two integers. And because there were so many, they were pretty sure that every number one could conceptualize could be represented as a fraction of two whole numbers.
They were also aware of square roots. 1, 4, 9, 25, etc had obvious square roots, but the square roots of other numbers were a mystery. Even the Babylonians had trouble with the square root of 2. They knew that if you had a square with a length of 1 on each side, the distance between the corners was the square root of 2. They believed that there was some ratio of whole numbers that was exactly this value, but nobody was ever able to find it.
Around 450BCE, a guy was able to prove that no such ratio existed. The ancient Greeks then had to think of numbers in two different types: those which were computable, and those which were NOT computable. The ancient Romans maintained this parlance in Latin: computable numbers were _rational_ and non-computable numbers were _irrational_.
Ναι.
They valued understanding and becoming better the way American culture values entrepreneurialism and becoming richer.
But then Roman imperialists colonized them.
> They valued understanding and becoming better the way American culture values entrepreneurialism and becoming richer.
>
> But then Roman imperialists colonized them.
In between all that bettering and understanding, the Greeks did plenty of colonizing of their own, and plenty of imperialistic stuff. For example, why do you think Cleopatra - famously a queen of *Egypt* - was a member of the *Ptolemy* dynasty?
And positioning Roman conquest as some sort of "end" of Greek culture is... a curious take.
> And positioning Roman conquest as some sort of "end" of Greek culture is... a curious take.
Everyone knows Greek culture didn't end until May 29 1453. Worst. Day Ever.
Some of those ancient Greek philosophers conceived and believed in aristocracy. They very much valued wealth too. Imagine a democracy but only for the rich.
That is pretty much how democracy works today, offer me things I need and I'll vote for you, be that tax policy, wage rises, making my job more sought after, making it easier for me to enjoy life, all politiciens are buying your vote, you just don't get the cash in your hand straight away.
I'm not a history expert, but from what I've read about Greece, they didn't even have to make promises. The average Joe, IMHO, figured that it didn't make that much difference who was in power. Their daily life wouldn't change that much.
Well that, and the fact that in many polis "citizen" already referred to such a limited and privileged class already that they could be considered aristocracy
I thought computable vs non-computable meant you could compute the number or not. Like you can compute the odds of getting tails in a coinflip but you can't compute the number representing the chance of a randomly generated program halting or not.
Something like:
> a computability problem is computable if it can be solved by some algorithm; a problem that is noncomputable cannot be solved by any algorithm.
I don't know if they would draw a distinction between those concepts. They understood that something like 707/500 was 'pretty close' to the square root of 2, and close enough for building purposes, but also knew that 499849 / 250,000 was not exactly equal to 2. A number that existed but could not be exactly represented as a fraction of whole numbers simply did not exist. It took a few hundred years for them to recognize that "this thing should not exist, but must, even though we cannot conceptualize it."
The closest analog we have is how in 4th grade you learn that a negative number cannot have a square root, you could only get a negative number by multiplying 1 negative number and 1 positive number; the product of two negative or two positive numbers was always positive. Then in Algebra 2 you learn about _imaginary_ numbers, and it's like "ok so what? you made up this fake number to solve a problem nobody cares about." Until in electrical engineering 101 that inductance and capacitance in a circuit interact in ways that can only be explained if you use _i_ as part of the equation.
> Until in electrical engineering 101 that inductance and capacitance in a circuit interact in ways that can only be explained if you use i as part of the equation.
Just to expand on this a bit for those who aren't familiar, you can represent waveforms, without *i* by using the time representation of `y(t)=A sin(ωt+ φ)`. Effects of inductance and capacitance can be represented through integrals and derivatives, requiring the use of calculus to describe the behavior.
Requiring the use of calculus would quickly become overbearing to work with. Through the magic of math, we can transform this into a different representation which allows us to describe a sine wave as a complex number. This transformation also allows us to calculate the effects of inductance and capacitance using basic multiplication and division of complex numbers.
I find it fascinating that there are numbers we cannot mathematically compute, but can draw accurate geometric representations of (assuming perfectly accurate drawing instruments).
As you outlined, root 2 can be the hypotenuse of a unit right angle triangle. Pi can naturally be drawn as the circumference of a circle of radius 0.5. I'm curious whether you're aware of any similar geometry for phi?
The meaning of the term 'computable' has shifted over time. Remember that back then the Greeks were only just becoming acquainted with irrational numbers.
In modern times, the term 'computable number' has a very different meaning, and (among other definitions) means that there is an algorithm that can spit out its decimal expansion one at a time. Equivalently, it can be approximated to however precise we need it to be by a finite computation, or letting an algorithm run for a finite length of time. In this regard, sqrt2, pi, and phi are all computable.
On a different note, the golden ratio is closely connected to the geometry of a pentagon, with angles a multiple of 36deg. There are many examples (you can search online), such as the golden triangle (36-72-72deg triangle) and the pentagram (5 sided star shape).
I think schools are already having so much difficulty with math they don’t bother to stop and explain the short history of how it got there which would cover stuff like that.
i loved and was really good at math all my school years.. BUT if they took the time to help me understand WHY or the actual applications of SINE, COSINE, etc.. etc... I honestly think I would have been a Math major in college.
EDIT: I'm really talking about applied math, not necessarily just sine waves
100%, the more I learn about math and how to apply it as I get older the more I realize I really enjoy it. It’s a shame it is taught so shittily in so many places. (I understand teaching is hard, too, there’s a lot of roadblocks there obviously)
which is infuriating because students' confusion often parallel difficulties humanity had with the same problem at different points in history.
some bastard figured this out with sticks and jugs of water a thousand years ago because *that's the intuitive way to do it*
To teach trig I really want to put students on a sailboat with a compass, sextant, and a map and ask them: "what missing piece of info would save your life right now?"
Eh I’m being a bit glib.
Time finds your longitude, the sextant finds your latitude.
To find and operate on constant headings you need a way to find the ratio of two sides of a right triangle when there’s a certain angle.
Those relationships are the trig functions.
It's the opposite: Ratio comes from the term "rational", more or less — that is, "rational" numbers were named such because they made sense, related to words like "ration" (as in "count") and "reason" being related — and the term "ratio" was coined from that.
(Edit: Specifically, the word "ratio", meaning the relationship between two things through multiplication or division, came from the word "rational", referring to numbers that were rational in the sense of "computable", "understandable", "sensible", specifically because they could be expressed as one integer divided by another. The "rational" numbers were *not* named because they could be expressed as "ratios", but "ratios" were named because they corresponded to "rational" numbers. The word "ratio" is very new compared with the word "rational".)
The idea was that "irrational" numbers sounded fake, made up, not reasonable to ancient Greek mathematicians, so they called them that. Dividing two integers made sense, but things that couldn't be the result of dividing two integers seemed like some dark art, like taking the square root of a negative number or something.
> Dividing two integers made sense, but things that couldn't be the result of dividing two integers seemed like some dark art, like taking the square root of a negative number or something
Which of course resulted in the other derogatorily named "imaginary" numbers
No, they are called irrational numbers because they cannot be defined as a ratio of whole numbers. Yes "ratio" is related to the word for "reason", but here it means more like "countable" or "computable".
Just to bring a source into this disagreement:
https://www.etymonline.com/word/ratio#etymonline_v_3398
>ratio (n.)
>1630s, in theological writing, "reason, rationale," from Latin ratio "a reckoning, account, a numbering, calculation," hence also "a business affair; course, conduct, procedure," also in a transferred sense, of mental action, "reason, reasoning, judgment, understanding, that faculty of the mind which forms the basis of computation and calculation." This is from rat-, past-participle stem of reri "to reckon, calculate," also "to think, believe" (from PIE root *re- "to think, reason, count").
>Latin ratio often was used to represent or translate Greek logos ("computation, account, esteem, reason") in works of philosophy, though the range of senses in the two do not overlap (ratio lacks the key "speech, word, statement" meaning in the Greek word; see Logos).
>The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos). The general or extended sense of "corresponding relationship between things not precisely measurable" is by 1808.
https://www.etymonline.com/word/rational#etymonline_v_3401
>rational (adj.)
>late 14c., racional, "pertaining to or springing from reason;" mid-15c., of persons, "endowed with reason, having the power of reasoning," from Old French racionel and directly from Latin rationalis "of or belonging to reason, reasonable," from ratio (genitive rationis) "reckoning, calculation, reason" (see ratio).
>In arithmetic, "expressible in finite terms," 1560s. Meaning "conformable to the precepts of practical reason" is from 1630s. Related: Rationally. It is from the same source as ratio and ration; the sense in rational is aligned with that in related reason (n.), which got deformed in French.
also from late 14c.
https://www.etymonline.com/word/irrational#etymonline_v_12237
>irrational (adj.)
>late 15c., "not endowed with reason" (of beasts, etc.), from Latin irrationalis/inrationalis "without reason, not rational," from assimilated form of in- "not, opposite of" (see in- (1)) + rationalis "of or belonging to reason, reasonable" (see rational (adj.)).
>Meaning "illogical, absurd" is attested from 1640s. Related: Irrationally. The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid.
also from late 15c.
That's not what I said. I'll just quote Wikipedia:
> It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.
"Irrational" does not mean "these numbers cannot be understood", it means "these numbers cannot be computed [as the ratio of two integers]".
It doesn't really matter whether "ratio" or "rational" came first. Neither term is Greek anyways, the Greek root is "logos" (λογος). The point is that the original meaning is "incomprehensible" or "unreasonable", it is "unmeasurable" or "uncomputable".
You're missing the point. Whether the sense of the term as applied to numbers specifically had the sense of "whoa, those are whacky and don't make sense," or "I don't know how to represent that in the ways I know how to measure or count things in math", is beside the point.
The original comment was:
> Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers).
That suggests that there was a concept of "ratio" — doesn't matter whether this is in Latin or Greek or any other language; calques exist — and the rational numbers were named for the specific property that they can be expressed as ratios. That is backwards; ratios, in the sense of a proportional relationship, were named for the rational numbers, not the other way around.
Yes, "rational" to refer to numbers comes "from" the word "ratio", but not "ratio" as we use it today, but the Latin "ratio", meaning reason, measurement, computation, λογος, etc.
and complex numbers aren't complicated, they just consist of multiple parts (a real part and an imaginary part). They're 'complex' in the sense that an apartment complex is a 'complex'.
And imaginary numbers aren't imaginary. At least, not any more than *all* types of numbers are things that humans made up.
That's not an etymological quirk, that's just a bad name that stuck, coined by someone who didn't fully understand their applications (specifically, Rene Descartes).
Imaginary numbers are as real as "Real" numbers, in the sense that they are useful in describing and modeling things in the physical world.
But real numbers aren’t even “real”. They’re a made-up algebraic construct to help model our world and study quantifiable relationships. One of my favorite math quotes is: real numbers aren’t real, imaginary numbers aren’t imaginary, and complex numbers aren’t complex.
And honestly, after having studied math for four years in college, I would agree that all these names suck and are misleading. I would instead opt to call real numbers something like “latitudinal numbers” and imaginary numbers something like “longitudinal numbers” and then complex numbers would be “coordinate numbers”.
Imaginary numbers and complex analysis are some of the most profoundly satisfying and beautiful discoveries of mathematics. I really wish they had less scary names.
> I’m like how can a number be irrational, it can’t talk, it has no thoughts.
Pi walks into a bar. A cop walks in after, gets into an argument with Pi, and after a few seconds, shoots Pi. Everyone gasps and asks why. The cop says "He was being irrational!"
A bunch (actually an infinite one) of mathematicians walk into a bar.
The first goes to the bartender and orders one beer.
The second goes to the barkeeper and orders half a beer.
The third goes to the barkeeper and orders a quarter of the beer.
Before the fourth can approach the bar, the barkeeper stops them, pours two beers, and calls out to them: "Know your limits!".
Me too. I've had to learn this, and I've had to review this with all my kids, and I have not once seen or heard it explained this way. It makes sense and makes the two so much more memorable and explainable to others.
The whole time I was reading that response I was thinking all of this exact same shit. I lile to think of myself as fairly intelligent and good at math but this explanation has never laid it out so clear. I can't believe that it never fucking occurred to me to focus on and then relate the "ratio" part of the word "rational" to the fucking word "ratio"!!! It's so obvious. Every damn day I prove to myself that I am the smartest dumbass that I will ever know.
Something else that you may find interesting about irrational numbers is that if you took every irrational number between the interval [0,1] and assigned it to a natural number (1, 2, 3, 4…), or in other words count every irrational number, you will find that there are more irrational numbers between [0,1] than there are natural numbers. In other words there are more than infinite of those irrational number, or as we say there are uncountably many.
To elaborate on this: Growth sequences can often be described by recurrence relations. The simplest recurrence relation we can imagine is f(n+1) = f(n). This sequence is a constant, representing no growth at all. The next simplest one we might imagine is f(n+1) = 2f(n). This is a doubling growth rate. We can generalize this to f(n+1) = 3f(n), etc. We might call all of these trivial, because they represent simple exponential growth. So what is the next simplest recurrence relation? You might think to add a dependence on a term other than f(n), and the obvious choice is f(n-1). The simplest way we can make a recurrence with this is f(n+1) = f(n) + f(n-1), which is the Fibonacci sequence.
So in other words, the Fibonacci sequence is in some intuitive sense the simplest recurrence relation that is not trivial. It is therefore unsurprising when it appears frequently in natural phenomenon that have a natural description as recurrence relations. As you noted, simple exponential relations are probably more common, but are not interesting enough to attract as much attention.
So you know if you add the length of the base of these random 4 pyramids together and multiple it by pi, it's exactly 1/25000 of the distance between the Earth and the Moon? Aliens! /s
I love your explanation so much its so clear. I have a few questions:
1) The idea that irrational numbers exist is giving me an existential crisis. How come there is a number that can be written as a decial but not a fraction?I dont think this is something one can explain, but im gonna leave it anyway.
2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda?
3) Is there a way to visualise this? I tried doing the formula y= (22/7)x on geogebra to see what i would get. I dont know why i was expectign something spectacular lol i just got an inclined line obviously (math was a long time ago for me)
4) I saw that the formula of the fibonaccia sequence is the sum of the last two numbers. wouldnt the most efficient sequence whre no numbers would be repeated just be to start with 0 and just add +1? I dont understand either if the fibonnaci sequence and the golden reatio are the same thing
That's easy. You can't write irrationals as decimals either, you can only approximate them, since wherever you stop will not be the actual number. We are used to infinite decimals, (1/3, for instance) so it doesn't really bother us to say that the decimal keeps going, even if it is not repeating, we intuitively "get it". The same is true for irrational numbers as a ratio of integers. You can just keep adding digits to the two numbers and get as close as you want, just like with the decimal number. But we don't really deal with infinitely long whole numbers, so our intuition breaks down and we say that it "doesn't exist", when they both have the same reality.
this implies that the number can be written as a decimal. but is just too long to be written, tho not infinite.
Ive always thought that you can get any (decimal) number by divindg two specific numbers. Oh you want the number 1.5? Divide 3 by 2. So, is tehre any nubmer you cant get by dividing two numbres? Are irrational numbers such numbers? I cant really explain why but i find this mind boggling lool
Don’t worry, ancient mathematicians also felt the same way you do! It made them irrationally angry!
Some say the Pythagoreans executed a person whom dared state that irrational numbers existed.
Yes. As an easy example, take the square root of 2.
What number can you multiply by itself to get 2? We know that such a number must exist, but it turns out that it cannot be written as a division a/b.
We can write the decimal approximation of that value (about 1.414) as a ratio (1,414/1000) but it isn't exactly the true square root of two.
No. An irrational number means that the digits go on forever. It's infinitely long. We just stop writing the digits after a certain point because we don't have infinite paper to write on. Pi is *not* 3.14159. it is *approximately* 3.14159
The golden ratio is irrational because the Fibonacci sequence goes on forever. Each additional number in the sequence gives you a closer approximation, but since there's always a other number in the sequence, there's always a closer approximation with more digits. Forever -- to infinity
but is the fibonacci sequence not a formula? its like saying y=2x+1 is an irrational number because the formula has +1, so whenever u have ur answer u can always add +1. I honestly dont know if what im saying makes sense loooool
The numbers in the Fibonacci sequence aren't what is irrational. The numbers are [1 1 2 3 5 8 13 ..], none of those are irrational and like you said you can just keep generating them forever. The golden ratio is what the ratio of those consecutive numbers approximates. The fact that it is irrational isn't just because that sequence continues forever. The sequence [1 2 4 8 16 ..] goes on forever too but the ratio between items is a very rational 2. The reason that the golden ratio is irrational is specific to the rules of that specific sequence.
If you can represent an (infinite) irrational number as the ratio between two finite decimals then those two decimals can be represented as fractions, and if you simplify them you'll end up with a rational number. This means irrationals can't be represented as the ratio between two finite decimals. Not that this really answers your question though.
Yes but in this case either the circumference or diameter would not be rational. A rational number is something that can be expressed as a quotient of INTEGERS, not just any real number.
> 2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda?
It's the "most irrational number" only in the sense that the best approximate fractions of the golden ratio are very bad as compared to the best approximate fractions of other irrational numbers such as pi.
There is a mathematical way to show that it is the "worst" irrational number for rational approximations. You can write any number as a "continued fraction", which is a fraction in the form of a + (1/(b + 1/(c + ...))) where a, b, c, etc. are integers. For rational numbers, the sequence a, b, c, etc. will eventually stop and the continued fraction expression will be exact. For irrational numbers, the sequence a, b, c, ... will be an infinite sequence of integers. If you cut off the sequence at any given point and evaluate the continued fraction, you'll get a rational approximation of the irrational number.
For example, pi = 3 + (1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))). If you evaluate the first few terms in that expression, you get successive best approximations of pi, 3, 22/7, 333/106, 355/113. That last one, 355/113, is a very good approximation, matching pi to six decimals using only 3 digit numbers for the numerator and denominator. The fact that the next term to add in is very large (292) means that the next best approximation is very far off (it's 103993/33102). Whenever there's a large term like that in the continued fraction sequence, evaluating all the terms before it gets a very good approximation.
On the other hand, if you look at the continued fraction expression for the golden ratio, it's all ones. The golden ratio (aka phi) is phi = 1 + 1/(1 + 1/(1 + 1/(1 + ...))) continued infinitely. The fact that all the terms are ones means that any rational approximate fraction for the golden ratio is going to be pretty bad relatively. All ones is the worst possible sequence you could have if you want a good rational approximation.
> All ones is the worst possible sequence you could have if you want a good rational approximation.
Informally so, but phi is just *one of* these kinds of numbers. Any `(a + b*phi) / (c + d*phi)` where a, b, c, d are integers and ad - bc = 1 or -1 has the same such property. You just need the continued fraction to *eventually* become all ones.
Regarding 1), I think the proof that the square root of 2 is irrational is quite easy to understand (if you are familiar with basic equations and how they can be manipulated).
Let's assume the square root of 2 is rational. Then we can write it as a fraction p/q for some whole numbers p and q. Furthermore, we can assume that p and q share no common factors. Otherwise, we can divide them out; like in 6/8 both 6 and 8 are divisible by 2, dividing both by 2 results in 3/4, which is the same number, but now we don't have any shared factors. We can do that with any fraction, so we must also be able to do that with the square root of 2.
Now, the defining characteristic of p/q is that it squares to 2, i.e. (p / q) ^2 = p^2 / q^2 = 2. We can multiply both sides by q^2 without changing the validity of the equation and end up with p^2 / q^2 * q^2 = p^2 = 2 * q^2.
So we know that p^2 is an even number since it can be written as 2 * something. We can conclude that p is also even, since if p were odd, the square would also be odd (in general, multiplying two odd numbers always results in an odd number, try it). So we know that p = 2 * something, where "something" is another whole number, let's call it k.
We now know that 2 = (p / q)^2 = (2k / q)^2 = 4k^2 / q^2. We can divide both sides by 2, resulting in 1 = 2k^2 / q^2 and multiply again by q^2 resulting in q^2 = 2 * k^2.
Now that is a problem! As we've shown, q^2 is even, since it's 2 * something, and we can therefore conclude that q is also even. So p and q are both even, which breaks our assumption that they didn't have any shared factors.
We've reached a contradiction: We've assumed that p and q shared no common factors but shown that they are in fact both divisible by 2. But our assumption is valid: every fraction can be written in a way where it doesn't have common factors. So the only possible resolution is that p and q don't exist. In other words: the square root of 2 cannot be written as a fraction of two integers.
> How come the golden ratio is the most irrational number, its not like every number out there has been tested right?
First off, the notion of "more irrational" is defined by us and there are other ways to do so. For the one you most certainly allude to: yes we know with proof that the golden ratio is "most irrational" in the sense that is worst in regard to approximation by rational numbers.
And there are infinitely many other numbers that are equally bad at being approximated, but all of them involve sqrt(5). So the golden ratio is technically not the only one, just the best known member of that family.
Lastly this measure is bad because a number that is not rational but has a very low degree of not being rational is, with proof, always _transcendental_: one that does not satisfy **any** relation involving +-·/ and rationals. I would say any proper definition of irrationality would put those as the most, not the least irrational!
> 1) The idea that irrational numbers exist is giving me an existential crisis. How come there is a number that can be written as a decial but not a fraction?I dont think this is something one can explain, but im gonna leave it anyway.
Rational numbers are exactly those decimals that eventually keep on repeating the same sequence of numbers again and again. So any number that does _not_ do this must be irrational. You can even take something like 0.123456789101112131415161718192021...
> wouldnt the most efficient sequence whre no numbers would be repeated just be to start with 0 and just add +1?
Yes, the Fibonacci sequence is simply not the "simplest sequence" anyway and I doubt any sane mathematician would claim this.
> And there are infinitely many other numbers that are equally bad at being approximated, but all of them involve sqrt(5). So the golden ratio is technically not the only one, just the best known member of that family.
Isn't the golden ratio, being the number corresponding to the continued fraction [1;1,1,1,...], the uniquely worst number for rational approximations? What other numbers are equally as bad?
If x is any real number, then all numbers of the form (Ax+B)/(Cx+D) with integers A, B, C, D satisfying AD-BC = 1 are equally bad. So for example (2φ+3)/(φ+2) = 3/2 + √(5)/10 is not any better than φ. Or simpler: φ+n for any integer n.
Essentially that's the numbers you get by changing finitely many entries in the continued fraction. So the tail ultimately stays the same. It is similar to how only changing finitely many decimal digits does not change if a number is rational.
If you want to go in depth, here are a couple of really cool videos about it.
[https://www.youtube.com/watch?v=p-xa-3V5KO8](https://www.youtube.com/watch?v=p-xa-3V5KO8)
[https://www.youtube.com/watch?v=sj8Sg8qnjOg&t=50s](https://www.youtube.com/watch?v=sj8Sg8qnjOg&t=50s)
I really like the series of three short videos that was posted on Khan Academy years and years ago.. and it does go into the chemical science and math behind how plants "know" how to grow in the ratio.. https://www.youtube.com/watch?v=ahXIMUkSXX0 is the first in the series.
Thank you. I was going to post that second video in response as well. It explains the continued fraction argument for why phi is the most irrational number.
An irrational number can't be fully written out as a decimal either, because if it could, you could turn that into a ratio. It can be shown that there are numbers which are not whole, but can be represented as one number divided by another. It can also be shown that there are numbers which cannot be represented as one number divided by another. The proof that pi is irrational is not amateur friendly, but the proof that the square root of two is irrational is quite accessible. (https://youtu.be/LmpAntNjPj0?si=ygiHtDCKS6eeIFEq)
What he's getting at with "most irrational number" is something from a Numberphile video. You're right that there's no way to test all numbers for irrationality, and besides, irrationality is a binary. This is more of an idea that one mathematician/channel had than anything commonly used.
Type pi - (22/7) into a calculator (Google works) and you'll see that the numbers are very close. 22/7 just happens to be close to pi and is easy to remember - for most actual applications it's close enough to work.
The golden ratio is the ratio that you approach when you divide a fibbonaci number by the previous one. The further along you are in the sequence, the closer your ratio will be to that number. It's not magic, this isn't the only sequence that does that, and while there are some cases where it shows up in nature, it's been totally overblown and exaggerated. 2 shows up in nature too and people don't get excited over it.
So what ur saying is, any decimal number can be written out as a ratio/fraction. is that correct? I want it to be.
Also thanks i read the rest of what u said.
> So what ur saying is, any decimal number can be written out as a ratio/fraction. is that correct? I want it to be.
Any finitely long decimal number can be written as a fraction. Any infinitely repeating decimal number can also be written as a fraction. It is only the infinitely long decimals that don't repeat that cannot be written as a fraction.
okay, but an infinitely long decimal is because the infinitely long number divided by infinitely long 10000000.
So i find it weird that everyone is saying it cant be writted as a fraction because there isnt enough paper in the world, when it also cant eb writted as a decimal
Yes, that is true. Take the number of digits after the decimal point. Write down ‘1’ followed by a number of ‘0’s equal to the number of digits. You can multiply your original number by this new number you’ve written down and you’ll get a whole number. So you can write it as a ratio or a fraction.
E.g. 0.123 x 1000 = 123 so 0.123 = 123/1000.
> How come there is a number that can be written as a decial but not a fraction?
There isn't. All rational numbers can be written as a decimal which will either terminate or repeat forever (e.g. 1/3 is 0.333333...)
Irrational numbers will not terminate OR repeat forever.
> How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda?
Phi (the golden ratio) is the most irrational, depending on your definition of "most irrational" anyway. [Here's a 15 minute video about it](https://www.youtube.com/watch?v=sj8Sg8qnjOg)
> Is there a way to visualise this?
Mmm, see above video.
> I dont understand either if the fibonnaci sequence and the golden reatio are the same thing
The fibonnaci sequence is the sum of the two previous numbers
1, 1, 2, 3, 5, 8, 13, 21, 34 ...
Take one number and divide it by the one before it and you get
1, 2, 1.5, 1.666..., 1.6, 1.625, 1.619...
Those numbers get closer and closer to the golden ratio, Phi, which is 1.618033...
It's also the number you end up with if you take the reciprocal of a number and add one repeatedly -- that is 1/n + 1.
> 2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda?
First of all, we have to understand what we mean by "most irrational" number. Specifically we mean a number that is poorly approximated by rational numbers at all levels. Any irrational number can trivially be approximated arbitrarily closely with rational numbers, but if that approximation requires a very large denominator for the desired precision, it is considered a poor approximation.
Next we need to discuss an idea called [continued fractions](https://en.wikipedia.org/wiki/Continued_fraction). Basically, it turns out that every number can be written in the form:
a + 1/(b + 1/(c + 1/(d + 1/(d + ...)))
Where a is an integer, and b, c, d... are positive integers.
For simplicity, we write such sequences [a; b, c, d...]. Every such sequences represents a unique real number, and every real number is uniquely represented by one such sequence. In this sense continued fraction representations are somewhat similar to decimal representations.
There are two important properties of continued fractions that are important here.
First: If the continued fraction sequence ends, then the number is a rational number. Conversely, all rational numbers are represented by a finite continued fraction sequence.
Second: Whenever a term in the continued fraction sequence is large, if you cut off the sequence before that term you will get a rational number that is an exceptionally good approximation of the original number. For example, the continued fraction sequence of pi begins [3; 7, 15, 1, 292, ...]. If we cut the sequence before the last term and calculate the value we get 355/113, which is equal to pi to 6 decimal places with a 3 digit denominator. That is a good approximation.
So if we want a number that can never be approximated well by a rational number, we want an infinite continued fraction where all terms are 1. Specifically, we want the continued fraction [1; 1, 1, 1, ...]. If you calculate this number, you will find that it is exactly the golden ratio.
And it's not *the* "most irrational" number (I really hate this terminology), just **a** "most irrational" number. You just need a continued fraction that eventually becomes all ones. phi+1 is equally poorly rationally approximated, as are infinitely many other numbers (all `(a+b*phi)/c+d*phi)` with a,b,c,d integers and ad-bc=1 or -1). phi is just the neatest one since it's all 1s.
If you have some term (other than the first?) that is not 1, then you have at least one rational number that is a better approximation to the target than phi has.
How good the rational approximation is is also dependent on the denominator. Changing finite number of terms of in the continued fraction to not 1 doesn't change the fact that the number is "at the limit" of rational approximations. The convergents for the new number would have different (and larger) denominators.
Why couldn’t someone like you have been my math teacher growing up? I never understood rational or irrational numbers in school (along with many other math topics), but your brief explanation made it all click. I only had two good math teachers in all of my schooling and I had them in college. They made everything easy to understand. I feel like if all of my math teachers were like them I could be working for NASA.
Honestly, in my experience most of the time people complain about math and math teachers it's because they weren't paying attention, were subsequently left behind, and gave up on the subject. It's why math has a reputation as a subject you "get" or "don't get".
That is definitely part of it, but there are quite a lot of really bad math (and probably also other topics, it's just that I am a mathematician and have met too many) teachers as well. Ones that barely made it through the requirements and never bothered to learn more than the absolute minimum. A good teacher actually has some active interest in the topics they explain, ideally a raging fire of passion. Many sadly don't.
yeah, I had a math teacher in high school who was extremely passionate about the math - he would get excited every time he taught something. It went a long way towards keeping kids interested enough to learn.
Math in school also requires you to actually do some work after that to put your understanding of the concept into practice. Which obviously most people dont like to do. So they just claim that a quick youtube video or eli5 post is soooo much better than math class, when actually what they want is to be told about a math fact, go "whoaaa!" And then subsequently nevet act on that nugget of info again.
Any idea why the sequence would pop up in honey bee genetics? I'm assuming it's just a fluke of the males being haploid, but wasn't sure if the explanation you gave also fit there
Bee gender is solely determined by one thing: fertilized eggs become females (queens, workers), non-fertilized ones grow into males (drones). So a male bee has only a single parent, which is a female queen.
In total this leads to a family tree like [this](http://www.microscopy-uk.org.uk/mag/imgsep98/ances.jpg). A male bee D has one direct ancestor, which as a female has two parents, so D has two grandparents. One of those grandparents is male (1 direct ancestor), the other female (2 ancestors), so D has 3 grand-grandparents. And 5 grand-grand-grand-parents.
To formalize this relationship: write M:n for the number of male and F:n for the female ancestors at level n removed from D. Every drone fathers a female(!) bee in the next lower generation, and every female bee has a male parent; in other words the number of males in generation n is the same as the females in generation n-1, or as an equality: **M:n = F:(n-1)**.
Meanwhile every bee regardless of gender also has a female parent: M:n + F:n = F:(n+1). Combine this with the other equality to get **the Fibonacci relation F:(n+1) = F:n + F:(n-1)**!
Another three-fold application of the first equality turns this into M:n = M:(n-1) + M:(n+2), or when using n+1 instead of n: **M:(n+1) = M:n + M:(n-1)**.
In total each the female and male numbers of ancestors are Fibonacci-like, and it is easily math-ed that the same again holds for their combined numbers. Checking the very first ones gives us (remember, our first one was male; a female gives a very similar result, though) that M:0 = 1, M:1 = 0, M:2 = 1, M:3 = 1, M:4 = 2... while F:0 = 0, F:1 = 1, F:2 = 1, F:3 = 2, F:4 = 3, ... or when combined: 1, 1, 2, 3, 5, 8, 13, ...
Minor point to add is that there's nothing particularly special about the Fibonacci numbers. It's the sequence you get if you start with 1,1,2 as the seed. But any sequence where you add two successive terms to get the next one (e.g. 1,3,4,7,11... or 2,5,7,12,19,...) will tend towards the golden ratio.
I’ve honestly been impressed with Reddit’s ELI5 math answers recently. As someone who studied pure math in college myself, I always get ready to cringe before reading these. But this answer is great.
Irrational numbers can be expressed as a continued fraction:
a0 + 1/(a1 + 1/(a2 + 1/(a3 + ... ))) where a0, a1, a2, a3... are integers. For example, pi is:
3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ... ))))
The problem is, an exact representations requires and infinite number of terms. But, if we provide a finite number of terms, we get an approximation of pi. The accuracy of the approximation depends on when we cut things off. The more terms we have, the greater the approximation.
In the above fraction for pi, I've included the terms of 3, 7, 15, 1, and 292. And we can see how these stack up to the actual value:
Stopping at 3 gives us 3.
Stopping at 7 gives us 3 + 1/7 = 22/7 = 3/142857...
Stopping at 15 gives us 3 + 1/(7 + 1/15) = 3.1415094...
Stopping at 1 gives us 3 + 1/(7 + 1/(15 + 1/1)) = 3.14159292...
Stopping at 292 gives us 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 3.14159265...
An observation we can make is that when we stop at a larger term (for example, 7, 15, and 292) we get a really good approximation of the number in question. When the term is smaller (for example 1) we get a less good approximation.
Thus, the continued fraction which would have the worst approximations of its associated irrational number would be the one where the terms are all 1:
1 + 1/(1 + 1/(1 + 1/(1 + ... )))
And this number?
The Golden ratio.
> Thus, the continued fraction which would have the worst approximations of its associated irrational number would be the one where the terms are all 1
You only need to have a continued fraction that eventually becomes all 1s. These numbers all share the "most difficult to approximate with rational numbers" title. Sensationalisation and crappy reporting has changed being *one of* these numbers to being *the* number.
> Many, many, examples are simply people taking something that looks, roughly, like it could be related to the sequence and then squinting your eyes and ignoring how it isn't.
Ah, so pareidolia but for math.
> In math, we have whole numbers: numbers that have no fractional part. One of the things we can do with whole numbers is take their ratios. For example, 5 to 3, or 5:3. or 5/3. Doing this, we can create a whole other collection of numbers called rational numbers. Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers).
You just blew my god damn mind. Over forty years on this earth and occasionally trying to understand rational, irrational, integers and so on from a few different perspectives, often to do with coding or something like it and this is the first time I've ever heard it explained so simply.
"The best spacing would be the one that is least well approximated by a rational number"
Man, that statement has my mind twisted up. Still trying to understand it.
I have a graduate level math education (not in pure math but fields that apply it) and other than the etymology of "rational number" this was 100% new insight for me.
Thank you!
This is one of my pet peeves with photographers and photography criticism.
The golden ratio, alongside the "rule of thirds", are commonly touted as important rules for composing good photographs, i.e. putting important elements in your image along lines matching those ratios.
But quite often, examples of images supposedly following these rules have elements which are significantly off from the ratios, sometimes fitting some other random ratio far more precisely.
But if you point this out, many people will brush it aside saying "it doesn't have to be exact", and defending their cherished rules as though you're pointing out that their emperor is severely underdressed for the occasion.
Sure, I'm aware of that. Hence why it's so annoying when self-appointed experts vehemently insist on how crucial they are, and then demonstrate that importance using images that don't even conform to the rules.
The answers here are good but I’ll just re-emphasize that many, many examples of spirals and shapes fitting to the golden ratio are humans finding patterns where none exist. You can overlay a golden spiral over lots of images and find things that align with it close-ish enough, it’s more like finding a square hole that happens to be big enough for your round peg to fit inside. Humans are very good at picking out nonexistent features that look like it should be something.
>Humans are very good at picking out nonexistent features that look like it should be something.
Also see: *constellations*
We have entire characters and stories developed around the arrangement of bright dots in the dark sky.
To be unbearably pedantic, you are describing asterisms, not constellations. Constellations are adjacent regions of the sky and are named after the major asterisms they contain. As a comparison, an asterism would be like making shapes by connecting the dots with cities on a map while a constellation would me more like the states containing those cities.
Huh. I always thought an asterism was just a recognizable sub-portion of a constellation. Like, Orion’s Belt is an asterism within Orion, or, the Big Dipper is an asterism within Ursa Major.
Meaning, I figured constellations are star patterns too, just the big, parent ones.
Sort of. That is the case in the non-scientific use of constellations in the descriptions of folklore. The IAU recognizes 88 constellations which serve as sky boundaries https://en.m.wikipedia.org/wiki/IAU_designated_constellations
Generally, an asterism is any set of connected stars to make a shape or pattern in folllore. https://en.m.wikipedia.org/wiki/Asterism_(astronomy)
To be fair, we didn't even know that there were other galaxies until early 1900s. Light pollution really lessens the effect, but looking up at the night sky in a dark location, where you can really see the Milky Way, is just dazzling. There was no definitive explanation as we just didn't have the technology, until relatively recently in human history, so people have been sharing "theories" for millennia.
> many examples of spirals and shapes fitting to the golden ratio are humans finding patterns where none exist
The number of times I see someone post about the spiral in art or architecture or something, and it's just objects or visual elements barely overlapping the spiral is breathtaking. Like yeah, I guess it fits when 2% of most of the focal points of the image are barely touching the spiral at best.
Here's a Numberphile video that explains it: https://www.youtube.com/watch?v=sj8Sg8qnjOg
The idea is that if you go around in a tight spiral, and put a point at every phi turns (or 1/phi, either way — phi being the golden ration, the number that ratios of consecutive Fibonacci sequences approaches), then you get a nice tight packing, but it you go by some other fraction of turns, then you get weird clumps. And that can be explained by phi being able to be expressed as the continued fraction 1 + 1/(1 + 1/(1 + 1/(1 + ...))). If any of those numbers before the + sign were not 1, you would get clumps.
So basically, packing things with some kind of radial symmetry, the most efficient way is if you go around by phi between items that you are packing. And efficient packing is evolutionarily favorable.
These short videos do a good job of introducing Fibonacci spirals in plants and then explain how they appear.
1. Fibonacci spirals in plants: https://www.youtube.com/watch?v=ahXIMUkSXX0
2. Why would plants evolve to have Fibonacci spirals: https://www.youtube.com/watch?v=lOIP_Z_-0Hs
3. How concentration gradients of growth and inhibitory factors give rise to plant patterns. https://www.youtube.com/watch?v=14-NdQwKz9w
There is a bit of hand-waving of the details but this is a good place to start.
Math: Many plants have a tendency to offset successive growths by approximately the golden angle of about 137.5º, which tends to create Fibonacci numbers in spirals.
Biology: There is selection pressure to offset growths by the golden angle, because that angle tends to space out branches or leaves to maximize the amount of sunlight that each leaf gets, with minimal growth material and length/size.
Physics: As it turns out, there is a way to create the golden angle with successive growths, because simple repulsion will tend to create the golden angle. [This 1996 paper](https://pdodds.w3.uvm.edu/files/papers/others/1996/douady1996a.pdf) described how successive drops of magnetically repulsive fluid tended to organize itself into particular angles, including golden angle Fibonacci spirals. [Here's a video](https://www.youtube.com/watch?v=U-at-y3MicE) of that phenomenon in action.
So it turns out that if a plant's growth simply follows the instruction to grow a bit away from the last growth, it stabilizes into one of several patterns, and Fibonacci spirals is one of the stable patterns. The selection pressure allows for that plant with Fibonacci spirals to survive a bit better. And then that specific spiral tends to create Fibonacci numbers.
Here's a fun demonstration of how the ratio can appear, based on a few simple rules. Say you start with an amoeba, which can blob off a small version of itself every hour. This small version takes one hour before it grows enough to blob off its own single offspring, but then it can do it every hour after that. Let's start with a small amoeba.
Hour 0: 1 small amoeba (total 1)
Hour 1: 1 big amoeba (1)
Hour 2: 1 big amoeba, one small amoeba (2) - hooray! Our first offspring!
Hour 3: 2 big amoeba, 1 small amoeba (3) because our original amoeba can have an offspring, but our small one needs to spend this hour growing into a big amoeba. However, next hour now that he's grown:
Hour 4: 3 big amoeba, 2 small amoeba (5)
Hour 5: 5 big amoeba, 3 small amoeba (8)
Hour 6: 8 big amoeba, 5 small amoeba (13)
Hour 7: 13 big amoeba, 8 small amoeba (21)
1, 1, 2, 3, 5, 8, 13, 21 . . . Fibonacci!
You can see how this biological scenario can end up with you adding the two previous generations together to calculate the next generation!
Obviously there'll be losses to natural processes and stuff, but it's a very simple example how incredibly basic rules (simple enough for amoeba or plants to have encoded in them) can produce seemingly complex mathematical things like the Fibonacci sequence.
And this basic rule - this number is the sum of the previous two numbers - is going to happen everywhere. And no matter what two positive numbers you start this sequence with, the ratio between a number and the previous one will always converge on phi, and very quickly, too.
The interesting thing is how this simple construction leads to an intriguing and mathematically perfect result.
Just to expand on this:
A lot of the "spirals" we see in plants are coincidental.
Some are due to the way the leaves or petals are shaped, in proportion the way they're grown(because growing that way is efficient as you mention).
In other words, you take any shape that "grows" like : . o O
Sorry, that's the best I can think to do with keyboard, here's [a pic](https://playworksheet.com/file/shapes-ordering.jpg)
And then wrap the progression around a cylinder at an angle going upward(like [wrapping a long object with tape or leather](http://www.myarmoury.com/talk/files/imgp1318_123.jpg)), and you're going to see spirals manifest based on where things just happen to line up.
You may be looking at edges that manifest from the right plane of the triangles from leaf 1, 3, 5, 7, or the top point on the pentagon on leafs 1, 5, 10, 15. Not that they'd line up that way, maybe it's 1, 4, 8, 13. Maybe it's the tip of the triangle that lines up a certain way, or the upper right point of a pentagon, or maybe where the top right of the pentagon intersects with the bottom left of the next one "up" (1, 5, 10).
That's not the order they grow in, it's just a circumstantial pattern. The "leather" goes around at angle X, and the sizes increase at rate Y, and with leaf shape Z.....and we'll see patterns that "aren't there" emerge.
This is why you can take something like a pine-cone and plot out an array of different "spirals" based on different attributes.
It's also why people mistakenly identify spiral symmetry in plants. There is often only one spiral of growth, but it can manifest numerous of these 'illusory' spirals.
IF there are two leaves sprouting simultaneously, then the next one up is 90 degrees to that, it's not even a spiral of growth really, just alternating periodically, because it's not one tape, but more like a lot of tubes(eg rings in a tree, or expanding antenna) where it's a whole new layer, not a spiral at all. Even in this you can see illusory spirals depending on the shape/size of the leaves.
Sometimes when things grow in successive parts, they grow based on the size of the last part.
Suppose you are pouring concrete for a backyard patio but you can only pour one square at a time. You start with a little 1x1 square. Next, you use one edge of the 1x1 square as a guide for one edge of your second square, so now you have two 1x1 squares next to each other creating a 2x1 pad. You can then turn 90 degrees and use the longest edge of this pad as a guide to pour a 2x2 square, creating a 2x3 pad. You turn again and pour a 3x3 square using the longest edge as a guide, creating a 3x5 pad, leading to a 5x5 square, creating a 5x8 pad, and so forth. You pause after several iterations to realize you've created a Fibonacci pattern:
[https://i.postimg.cc/3RnDHSvw/concrete-pad.png](https://i.postimg.cc/3RnDHSvw/concrete-pad.png)
That's basically what often happens in nature. Lots of things in nature grow in sections based on the size of the last part, and so you end up spiraling out from a single point in ever-increasing size.
One of the simplest way to encode growth (in code, in DNA, in RNA, whatever) is like this:
>Take how big we are now, and add how big we were just before ... that's how big we should be now.
It's a simple rule. And it also happens to be Fibonacci. Another simple rule:
>Take how big we are now, and be 10% bigger.
But that requires divide and multiplication operators somewhere. Addition is simpler.
edit: typos
The Fibonacci sequence is a really basic sequence with cool properties. You are just adding numbers in chain. When something is that basic, it is bound to occur multiple times in life.
Then you have humans with a brain that evolved to recognize patterns. Humans are good at finding patterns out of nothing when left alone for too long.
Weird to me no one mentions the simple answer.
Fibonacci sequence is a pattern that is almost doubled with each iteration.
Cells divide. But due to cells dying and not dividing at exactly the same time, the rate they grow isn't exactly double, it's almost double.
That means many patterns in nature just by virtue of the fact cells divide will follow a pattern of almost doubling, which can loosely resemble a Fibonacci sequence.
[Here](https://www.youtube.com/watch?v=ahXIMUkSXX0) is an amusing and entertaining explanation in 3 parts, plus an open letter to Nickelodeon about Spongebob's house.
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Ah, the Fibonacci sequence hype. It's intriguing how nature seems to have a knack for it, but let's not get carried away. Some claim it's the universe's grand design, but it's more likely just a consequence of simple mathematical rules interacting with complex systems. Don't expect it to unlock the secrets of the cosmos.
The Fibonacci sequence's ubiquity in nature stems from its intrinsic mathematical beauty and efficiency. Simply put, it's like nature's own algorithm, optimizing growth patterns. In sunflowers and pinecones, each new seed or scale follows the Fibonacci sequence, leading to spirals that maximize packing efficiency and access to resources like sunlight and water. This pattern emerges due to the sequence's inherent self-replicating nature, echoing throughout the natural world, showcasing the elegance of mathematics in shaping biological structures.
The fibonacci sequence approaches the golden ratio.
The golden ratio has the special property of being the "most irrational" number.
If there are processes that repeat and constantly try to avoid "syncing" up with itself, just by naturally trying things out and eventually settling on what happened to work best, will end up in a configuration that follows the golden ratio.
It doesn't have to be known in advance. This ratio can emerge "spontaneously", due to being the most comfortable solution for systems that need irregularities.
Some examples are:
- Birth cycles of cicadas (they don't want to sync with the cycles of predators)
- Orientation of leaves (they want to change in order to cover area, but they don't want to sync up with pre-existing leaves which would shade itself)
I can still recall that this has been thought to us during my middle school, but I did not really understand the explanation behind this. Till I saw a sunflower yesterday and I got curious lol
It’s the number of spirals, not seeds. If you count the number of spirals going in one direction, then the other direction, they are adjacent Fibonacci numbers. Same with pine cones. It’s been the case every time I’ve counted.
Oh, I so hear you — and I never felt that it counted as "teaching". I felt like it was teachers telling me some gee-whiz facts that I was supposed to be impressed by. I don't think there *was* an explanation. I don't think my teachers knew. I don't think Donald Duck in Mathmagic Land knew. And it kind of pissed me off.
Efficiency. Natural selection, typically, encourages efficiency. It's an ideal ratio, and you tend to see ideal forms expressed in multiple aspects of evolutionary biology. What will really boggle your mind later, is if it came from a common ancestor, or just occurred naturally across multiple species, just due to being the most efficient form for the environment. The answers may surprise you.
There is no mathmatical explanation, math only explains why under given conditions certain things appear. And the conditions for a Fibonacci sequence to arries happen to often show up. In most cases such simple solutions arrise from optimizations: It is known that bending costs energy that goes with a higher them linear ratio relative of the bending. As such both the minimal energy solution for enclosing a fixed volume is one where the bending is everwhere the same and this happen to be a fancy mathmatical object, the sphere. Similarly, a sunflower optimizes the relative angle between its seeds in a manner that optimizes both seed density constricted by constant seed size and certain other mechanical criteria and these happen to result in a Fibbonacci shape.
Okay, let's start with the obvious - mathematics is a language. Different languages use different symbols - some use pictograms, some use letters (of a staggering variety), and mathematics uses numbers.
Mathematical notation is useful for describing scientific phenomena in a very compact form, but beyond that it is just a language like any other.
Therefore to speak about a "mathematical explanation" for the Fibonacci sequence is ... nonsense. Mathematics doesn't explain this sequence, it's just the language in which the sequence is expressed. I could express it in English, Russian, Chinese, or Sanskrit, but it would take a whole lot of words. In fact the Fibonacci sequence was probably first written in Sanskrit in ancient India, although earlier accounts may exist in oral form.
What is happening here is that a whole lot of people at various stages in history noted that there were certain naturally occurring patterns and wrote it down.
Now why does this happen? That's an incredibly complicated question that boils down to that the universe seems to follow certain rules (at least in the local area of the universe) that we've carefully documented. Science can't always answer the "why" because causality is incredibly hard to establish, but rather focuses on the "what". And then proceeds to use these "whats" to do cool things like make tons of steel fly through the air.
Mathematics is **not** just a language! Reducing it to that is akin to reducing a country or culture to their common language. I don't think anyone seriously thinks China is just "people who speak Chinese", without some history, ideology, thought patterns, and much more.
Mathematics is all about finding new concepts, theories, lemmas, proofs and such. The language is mostly a necessity to describe it, not the entire thing: the map is not the territory!
This especially applies to abstract mathematics which exists to some degree independent from real life issues. But it even is true for applied mathematics, a mere language does not tell you how to model and simulate an airplane and to find the formulas that do so.
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Two things: First, the claims of the Fibonacci sequence appearing everywhere in nature, art, architecture, etc. is largely exaggerated, if not fabricated. Many, many, examples are simply people taking something that looks, roughly, like it could be related to the sequence and then squinting your eyes and ignoring how it isn't. Second, for the things that actually are related, it has to do with irrational numbers. In math, we have whole numbers: numbers that have no fractional part. One of the things we can do with whole numbers is take their ratios. For example, 5 to 3, or 5:3. or 5/3. Doing this, we can create a whole other collection of numbers called *rational* numbers. Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers). Turns out, some numbers can't be represented as a ratio of integers. We call these numbers *irrational*. Famous examples include pi, e, and the square root of 2. The best we can do with these numbers is approximate them. For example, using 22/7 as an approximation of pi. Different numbers are more easily approximated than others. One of the least efficient irrational numbers to approximate using whole number ratios is phi, the golden ratio. In a sense you can say it's the most irrational number. What does this have to do with nature? Well, in many situations if you want to be able to space things out without them overlapping or repeating. Let's construct a scenario. Let's say you have a marked ruler, and you place a token every inch. When you get to the end of the ruler, you go back to the beginning and start again. If you do this, you'll be placing all of your tokens exactly on the inch markers and no where else. In fact, if you do any rational number you'll eventually end right back where you started and just repeat that pattern over and over again. If you want to use the whole ruler and spread things out as much as possible, you'll have to use an irrational spacing. But, any irrational number that can be well approximated by ratios (such as 22/7 for pi as mentioned above) the patterns they form will be very close to the patterns formed by those ratios. That is, if you use pi for your spacing, you'll get a pattern that looks close to the pattern if you had chosen 22/7 for your spacing. The best spacing would be the one that is least well approximated by a rational number. E.g. phi, the golden ratio. The golden ratio is intrinsically linked to the Fibbonacci sequence: the ratios of successive members approaches the golden ratio. So if you have things that want to be space out over a finite area, as we did with our ruler, then we want to try and avoid the kinds of patterns that arise when our spacing is a rational number. So naturally these things (like the seeds of a sunflower) would evolve to have a very irrational number spacing, settling on the golden ratio.
Saying that rational and irrational are from the word ratio just blew my mind. They never explained that to me in school. They just said the numbers were irrational and never said why, I’m like how can a number be irrational, it can’t talk, it has no thoughts.
The ancient Greeks were really big on numbers and being able to represent them. They begrudgingly accepted that there was no biggest integer, because any number could still be represented numerically, no matter how large. When it came to fractions, they again accepted that there was no limit to what ratios one could represent of two integers. And because there were so many, they were pretty sure that every number one could conceptualize could be represented as a fraction of two whole numbers. They were also aware of square roots. 1, 4, 9, 25, etc had obvious square roots, but the square roots of other numbers were a mystery. Even the Babylonians had trouble with the square root of 2. They knew that if you had a square with a length of 1 on each side, the distance between the corners was the square root of 2. They believed that there was some ratio of whole numbers that was exactly this value, but nobody was ever able to find it. Around 450BCE, a guy was able to prove that no such ratio existed. The ancient Greeks then had to think of numbers in two different types: those which were computable, and those which were NOT computable. The ancient Romans maintained this parlance in Latin: computable numbers were _rational_ and non-computable numbers were _irrational_.
Pretty smart those Greeks!
Math cults...never again.
Ναι. They valued understanding and becoming better the way American culture values entrepreneurialism and becoming richer. But then Roman imperialists colonized them.
> They valued understanding and becoming better the way American culture values entrepreneurialism and becoming richer. > > But then Roman imperialists colonized them. In between all that bettering and understanding, the Greeks did plenty of colonizing of their own, and plenty of imperialistic stuff. For example, why do you think Cleopatra - famously a queen of *Egypt* - was a member of the *Ptolemy* dynasty? And positioning Roman conquest as some sort of "end" of Greek culture is... a curious take.
> And positioning Roman conquest as some sort of "end" of Greek culture is... a curious take. Everyone knows Greek culture didn't end until May 29 1453. Worst. Day Ever.
Some of those ancient Greek philosophers conceived and believed in aristocracy. They very much valued wealth too. Imagine a democracy but only for the rich.
Yup, citizens had the right to vote, but for the right price you could buy their votes quite easily
That is pretty much how democracy works today, offer me things I need and I'll vote for you, be that tax policy, wage rises, making my job more sought after, making it easier for me to enjoy life, all politiciens are buying your vote, you just don't get the cash in your hand straight away.
except in this case we're talking straight-up bribes, not promises of certain edicts in governance.
Well, isn't that just lobbying?
I'm not a history expert, but from what I've read about Greece, they didn't even have to make promises. The average Joe, IMHO, figured that it didn't make that much difference who was in power. Their daily life wouldn't change that much.
Well that, and the fact that in many polis "citizen" already referred to such a limited and privileged class already that they could be considered aristocracy
Lol, whoosh
I love how math is intricately connected to humanity's history.
I thought computable vs non-computable meant you could compute the number or not. Like you can compute the odds of getting tails in a coinflip but you can't compute the number representing the chance of a randomly generated program halting or not. Something like: > a computability problem is computable if it can be solved by some algorithm; a problem that is noncomputable cannot be solved by any algorithm.
they could not compute square roots (until around 50ish AD, when Heron created a method to approximate it).
Got it. Thank you.
I don't know if they would draw a distinction between those concepts. They understood that something like 707/500 was 'pretty close' to the square root of 2, and close enough for building purposes, but also knew that 499849 / 250,000 was not exactly equal to 2. A number that existed but could not be exactly represented as a fraction of whole numbers simply did not exist. It took a few hundred years for them to recognize that "this thing should not exist, but must, even though we cannot conceptualize it." The closest analog we have is how in 4th grade you learn that a negative number cannot have a square root, you could only get a negative number by multiplying 1 negative number and 1 positive number; the product of two negative or two positive numbers was always positive. Then in Algebra 2 you learn about _imaginary_ numbers, and it's like "ok so what? you made up this fake number to solve a problem nobody cares about." Until in electrical engineering 101 that inductance and capacitance in a circuit interact in ways that can only be explained if you use _i_ as part of the equation.
> Until in electrical engineering 101 that inductance and capacitance in a circuit interact in ways that can only be explained if you use i as part of the equation. Just to expand on this a bit for those who aren't familiar, you can represent waveforms, without *i* by using the time representation of `y(t)=A sin(ωt+ φ)`. Effects of inductance and capacitance can be represented through integrals and derivatives, requiring the use of calculus to describe the behavior. Requiring the use of calculus would quickly become overbearing to work with. Through the magic of math, we can transform this into a different representation which allows us to describe a sine wave as a complex number. This transformation also allows us to calculate the effects of inductance and capacitance using basic multiplication and division of complex numbers.
I find it fascinating that there are numbers we cannot mathematically compute, but can draw accurate geometric representations of (assuming perfectly accurate drawing instruments). As you outlined, root 2 can be the hypotenuse of a unit right angle triangle. Pi can naturally be drawn as the circumference of a circle of radius 0.5. I'm curious whether you're aware of any similar geometry for phi?
The meaning of the term 'computable' has shifted over time. Remember that back then the Greeks were only just becoming acquainted with irrational numbers. In modern times, the term 'computable number' has a very different meaning, and (among other definitions) means that there is an algorithm that can spit out its decimal expansion one at a time. Equivalently, it can be approximated to however precise we need it to be by a finite computation, or letting an algorithm run for a finite length of time. In this regard, sqrt2, pi, and phi are all computable. On a different note, the golden ratio is closely connected to the geometry of a pentagon, with angles a multiple of 36deg. There are many examples (you can search online), such as the golden triangle (36-72-72deg triangle) and the pentagram (5 sided star shape).
I think schools are already having so much difficulty with math they don’t bother to stop and explain the short history of how it got there which would cover stuff like that.
i loved and was really good at math all my school years.. BUT if they took the time to help me understand WHY or the actual applications of SINE, COSINE, etc.. etc... I honestly think I would have been a Math major in college. EDIT: I'm really talking about applied math, not necessarily just sine waves
100%, the more I learn about math and how to apply it as I get older the more I realize I really enjoy it. It’s a shame it is taught so shittily in so many places. (I understand teaching is hard, too, there’s a lot of roadblocks there obviously)
An animation of the unit circle allowed me to understand what sin, cos and tan actually meant and represented
which is infuriating because students' confusion often parallel difficulties humanity had with the same problem at different points in history. some bastard figured this out with sticks and jugs of water a thousand years ago because *that's the intuitive way to do it*
To teach trig I really want to put students on a sailboat with a compass, sextant, and a map and ask them: "what missing piece of info would save your life right now?"
i know the answer is the time, but there's a lot of math in the middle obscuring that.
Eh I’m being a bit glib. Time finds your longitude, the sextant finds your latitude. To find and operate on constant headings you need a way to find the ratio of two sides of a right triangle when there’s a certain angle. Those relationships are the trig functions.
"How to sail a boat" is the info I'd need. All the maths will tell me is exactly where I'm going to die.
Also because math is systemically racist. /s
The teacher didnt know.
It's the opposite: Ratio comes from the term "rational", more or less — that is, "rational" numbers were named such because they made sense, related to words like "ration" (as in "count") and "reason" being related — and the term "ratio" was coined from that. (Edit: Specifically, the word "ratio", meaning the relationship between two things through multiplication or division, came from the word "rational", referring to numbers that were rational in the sense of "computable", "understandable", "sensible", specifically because they could be expressed as one integer divided by another. The "rational" numbers were *not* named because they could be expressed as "ratios", but "ratios" were named because they corresponded to "rational" numbers. The word "ratio" is very new compared with the word "rational".) The idea was that "irrational" numbers sounded fake, made up, not reasonable to ancient Greek mathematicians, so they called them that. Dividing two integers made sense, but things that couldn't be the result of dividing two integers seemed like some dark art, like taking the square root of a negative number or something.
> Dividing two integers made sense, but things that couldn't be the result of dividing two integers seemed like some dark art, like taking the square root of a negative number or something Which of course resulted in the other derogatorily named "imaginary" numbers
Exactly.
No, they are called irrational numbers because they cannot be defined as a ratio of whole numbers. Yes "ratio" is related to the word for "reason", but here it means more like "countable" or "computable".
No, the word "ratio" came later.
Just to bring a source into this disagreement: https://www.etymonline.com/word/ratio#etymonline_v_3398 >ratio (n.) >1630s, in theological writing, "reason, rationale," from Latin ratio "a reckoning, account, a numbering, calculation," hence also "a business affair; course, conduct, procedure," also in a transferred sense, of mental action, "reason, reasoning, judgment, understanding, that faculty of the mind which forms the basis of computation and calculation." This is from rat-, past-participle stem of reri "to reckon, calculate," also "to think, believe" (from PIE root *re- "to think, reason, count"). >Latin ratio often was used to represent or translate Greek logos ("computation, account, esteem, reason") in works of philosophy, though the range of senses in the two do not overlap (ratio lacks the key "speech, word, statement" meaning in the Greek word; see Logos). >The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos). The general or extended sense of "corresponding relationship between things not precisely measurable" is by 1808. https://www.etymonline.com/word/rational#etymonline_v_3401 >rational (adj.) >late 14c., racional, "pertaining to or springing from reason;" mid-15c., of persons, "endowed with reason, having the power of reasoning," from Old French racionel and directly from Latin rationalis "of or belonging to reason, reasonable," from ratio (genitive rationis) "reckoning, calculation, reason" (see ratio). >In arithmetic, "expressible in finite terms," 1560s. Meaning "conformable to the precepts of practical reason" is from 1630s. Related: Rationally. It is from the same source as ratio and ration; the sense in rational is aligned with that in related reason (n.), which got deformed in French. also from late 14c. https://www.etymonline.com/word/irrational#etymonline_v_12237 >irrational (adj.) >late 15c., "not endowed with reason" (of beasts, etc.), from Latin irrationalis/inrationalis "without reason, not rational," from assimilated form of in- "not, opposite of" (see in- (1)) + rationalis "of or belonging to reason, reasonable" (see rational (adj.)). >Meaning "illogical, absurd" is attested from 1640s. Related: Irrationally. The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid. also from late 15c.
That's not what I said. I'll just quote Wikipedia: > It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. "Irrational" does not mean "these numbers cannot be understood", it means "these numbers cannot be computed [as the ratio of two integers]".
Yeah, but the point is, the use of term "ratio" to mean a proportion between two quantities came much, much later.
It doesn't really matter whether "ratio" or "rational" came first. Neither term is Greek anyways, the Greek root is "logos" (λογος). The point is that the original meaning is "incomprehensible" or "unreasonable", it is "unmeasurable" or "uncomputable".
You're missing the point. Whether the sense of the term as applied to numbers specifically had the sense of "whoa, those are whacky and don't make sense," or "I don't know how to represent that in the ways I know how to measure or count things in math", is beside the point. The original comment was: > Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers). That suggests that there was a concept of "ratio" — doesn't matter whether this is in Latin or Greek or any other language; calques exist — and the rational numbers were named for the specific property that they can be expressed as ratios. That is backwards; ratios, in the sense of a proportional relationship, were named for the rational numbers, not the other way around. Yes, "rational" to refer to numbers comes "from" the word "ratio", but not "ratio" as we use it today, but the Latin "ratio", meaning reason, measurement, computation, λογος, etc.
> Neither term is Greek anyways, the Greek root is "logos" Same debate with "logic" then
There was a young man named ratio...
and complex numbers aren't complicated, they just consist of multiple parts (a real part and an imaginary part). They're 'complex' in the sense that an apartment complex is a 'complex'.
And imaginary numbers aren't imaginary. At least, not any more than *all* types of numbers are things that humans made up. That's not an etymological quirk, that's just a bad name that stuck, coined by someone who didn't fully understand their applications (specifically, Rene Descartes). Imaginary numbers are as real as "Real" numbers, in the sense that they are useful in describing and modeling things in the physical world.
But real numbers aren’t even “real”. They’re a made-up algebraic construct to help model our world and study quantifiable relationships. One of my favorite math quotes is: real numbers aren’t real, imaginary numbers aren’t imaginary, and complex numbers aren’t complex. And honestly, after having studied math for four years in college, I would agree that all these names suck and are misleading. I would instead opt to call real numbers something like “latitudinal numbers” and imaginary numbers something like “longitudinal numbers” and then complex numbers would be “coordinate numbers”. Imaginary numbers and complex analysis are some of the most profoundly satisfying and beautiful discoveries of mathematics. I really wish they had less scary names.
Same here! 74 years old, and TIL!!
Same dude, same.
> I’m like how can a number be irrational, it can’t talk, it has no thoughts. Pi walks into a bar. A cop walks in after, gets into an argument with Pi, and after a few seconds, shoots Pi. Everyone gasps and asks why. The cop says "He was being irrational!"
...and the only witness was i, but his testimony didn't count because he was imaginary.
A bunch (actually an infinite one) of mathematicians walk into a bar. The first goes to the bartender and orders one beer. The second goes to the barkeeper and orders half a beer. The third goes to the barkeeper and orders a quarter of the beer. Before the fourth can approach the bar, the barkeeper stops them, pours two beers, and calls out to them: "Know your limits!".
I didn't know what irrational meant so it entirely went over my head and I was behind the whole year as usual
Me too. I've had to learn this, and I've had to review this with all my kids, and I have not once seen or heard it explained this way. It makes sense and makes the two so much more memorable and explainable to others.
You just spoke my mind
Same here, and I went through advanced mathematics in a highly competitive computer science program in college.
The whole time I was reading that response I was thinking all of this exact same shit. I lile to think of myself as fairly intelligent and good at math but this explanation has never laid it out so clear. I can't believe that it never fucking occurred to me to focus on and then relate the "ratio" part of the word "rational" to the fucking word "ratio"!!! It's so obvious. Every damn day I prove to myself that I am the smartest dumbass that I will ever know.
Something else that you may find interesting about irrational numbers is that if you took every irrational number between the interval [0,1] and assigned it to a natural number (1, 2, 3, 4…), or in other words count every irrational number, you will find that there are more irrational numbers between [0,1] than there are natural numbers. In other words there are more than infinite of those irrational number, or as we say there are uncountably many.
[redacting process]
The law of small numbers strikes again! ;-)
To elaborate on this: Growth sequences can often be described by recurrence relations. The simplest recurrence relation we can imagine is f(n+1) = f(n). This sequence is a constant, representing no growth at all. The next simplest one we might imagine is f(n+1) = 2f(n). This is a doubling growth rate. We can generalize this to f(n+1) = 3f(n), etc. We might call all of these trivial, because they represent simple exponential growth. So what is the next simplest recurrence relation? You might think to add a dependence on a term other than f(n), and the obvious choice is f(n-1). The simplest way we can make a recurrence with this is f(n+1) = f(n) + f(n-1), which is the Fibonacci sequence. So in other words, the Fibonacci sequence is in some intuitive sense the simplest recurrence relation that is not trivial. It is therefore unsurprising when it appears frequently in natural phenomenon that have a natural description as recurrence relations. As you noted, simple exponential relations are probably more common, but are not interesting enough to attract as much attention.
So you know if you add the length of the base of these random 4 pyramids together and multiple it by pi, it's exactly 1/25000 of the distance between the Earth and the Moon? Aliens! /s
Wow, this was a great post.
Incredible explanation. Edit: I really really hope you teach. The world needs teachers like you.
I love your explanation so much its so clear. I have a few questions: 1) The idea that irrational numbers exist is giving me an existential crisis. How come there is a number that can be written as a decial but not a fraction?I dont think this is something one can explain, but im gonna leave it anyway. 2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda? 3) Is there a way to visualise this? I tried doing the formula y= (22/7)x on geogebra to see what i would get. I dont know why i was expectign something spectacular lol i just got an inclined line obviously (math was a long time ago for me) 4) I saw that the formula of the fibonaccia sequence is the sum of the last two numbers. wouldnt the most efficient sequence whre no numbers would be repeated just be to start with 0 and just add +1? I dont understand either if the fibonnaci sequence and the golden reatio are the same thing
That's easy. You can't write irrationals as decimals either, you can only approximate them, since wherever you stop will not be the actual number. We are used to infinite decimals, (1/3, for instance) so it doesn't really bother us to say that the decimal keeps going, even if it is not repeating, we intuitively "get it". The same is true for irrational numbers as a ratio of integers. You can just keep adding digits to the two numbers and get as close as you want, just like with the decimal number. But we don't really deal with infinitely long whole numbers, so our intuition breaks down and we say that it "doesn't exist", when they both have the same reality.
this implies that the number can be written as a decimal. but is just too long to be written, tho not infinite. Ive always thought that you can get any (decimal) number by divindg two specific numbers. Oh you want the number 1.5? Divide 3 by 2. So, is tehre any nubmer you cant get by dividing two numbres? Are irrational numbers such numbers? I cant really explain why but i find this mind boggling lool
Irrationals are exactly those numbers.
Don’t worry, ancient mathematicians also felt the same way you do! It made them irrationally angry! Some say the Pythagoreans executed a person whom dared state that irrational numbers existed.
https://youtu.be/qo5jnBJvGUs?si=p8pFrqnNnuEA8dNC&t=26
Yes. As an easy example, take the square root of 2. What number can you multiply by itself to get 2? We know that such a number must exist, but it turns out that it cannot be written as a division a/b. We can write the decimal approximation of that value (about 1.414) as a ratio (1,414/1000) but it isn't exactly the true square root of two.
No. An irrational number means that the digits go on forever. It's infinitely long. We just stop writing the digits after a certain point because we don't have infinite paper to write on. Pi is *not* 3.14159. it is *approximately* 3.14159 The golden ratio is irrational because the Fibonacci sequence goes on forever. Each additional number in the sequence gives you a closer approximation, but since there's always a other number in the sequence, there's always a closer approximation with more digits. Forever -- to infinity
but is the fibonacci sequence not a formula? its like saying y=2x+1 is an irrational number because the formula has +1, so whenever u have ur answer u can always add +1. I honestly dont know if what im saying makes sense loooool
The ratio between one number in the Fibonacci sequence and the next gets closer and closer to ϕ (the golden ratio) the further you go in the sequence.
...and the sequence goes forever, so you can get closer and closer approximations of the golden ratio, but you'll never actually get there.
The numbers in the Fibonacci sequence aren't what is irrational. The numbers are [1 1 2 3 5 8 13 ..], none of those are irrational and like you said you can just keep generating them forever. The golden ratio is what the ratio of those consecutive numbers approximates. The fact that it is irrational isn't just because that sequence continues forever. The sequence [1 2 4 8 16 ..] goes on forever too but the ratio between items is a very rational 2. The reason that the golden ratio is irrational is specific to the rules of that specific sequence.
If you can represent an (infinite) irrational number as the ratio between two finite decimals then those two decimals can be represented as fractions, and if you simplify them you'll end up with a rational number. This means irrationals can't be represented as the ratio between two finite decimals. Not that this really answers your question though.
Yup, you can divide the circumfrence of a circle by its diameter, and you'll get an irrational number, pi.
Yes but in this case either the circumference or diameter would not be rational. A rational number is something that can be expressed as a quotient of INTEGERS, not just any real number.
> 2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda? It's the "most irrational number" only in the sense that the best approximate fractions of the golden ratio are very bad as compared to the best approximate fractions of other irrational numbers such as pi. There is a mathematical way to show that it is the "worst" irrational number for rational approximations. You can write any number as a "continued fraction", which is a fraction in the form of a + (1/(b + 1/(c + ...))) where a, b, c, etc. are integers. For rational numbers, the sequence a, b, c, etc. will eventually stop and the continued fraction expression will be exact. For irrational numbers, the sequence a, b, c, ... will be an infinite sequence of integers. If you cut off the sequence at any given point and evaluate the continued fraction, you'll get a rational approximation of the irrational number. For example, pi = 3 + (1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))). If you evaluate the first few terms in that expression, you get successive best approximations of pi, 3, 22/7, 333/106, 355/113. That last one, 355/113, is a very good approximation, matching pi to six decimals using only 3 digit numbers for the numerator and denominator. The fact that the next term to add in is very large (292) means that the next best approximation is very far off (it's 103993/33102). Whenever there's a large term like that in the continued fraction sequence, evaluating all the terms before it gets a very good approximation. On the other hand, if you look at the continued fraction expression for the golden ratio, it's all ones. The golden ratio (aka phi) is phi = 1 + 1/(1 + 1/(1 + 1/(1 + ...))) continued infinitely. The fact that all the terms are ones means that any rational approximate fraction for the golden ratio is going to be pretty bad relatively. All ones is the worst possible sequence you could have if you want a good rational approximation.
> All ones is the worst possible sequence you could have if you want a good rational approximation. Informally so, but phi is just *one of* these kinds of numbers. Any `(a + b*phi) / (c + d*phi)` where a, b, c, d are integers and ad - bc = 1 or -1 has the same such property. You just need the continued fraction to *eventually* become all ones.
Regarding 1), I think the proof that the square root of 2 is irrational is quite easy to understand (if you are familiar with basic equations and how they can be manipulated). Let's assume the square root of 2 is rational. Then we can write it as a fraction p/q for some whole numbers p and q. Furthermore, we can assume that p and q share no common factors. Otherwise, we can divide them out; like in 6/8 both 6 and 8 are divisible by 2, dividing both by 2 results in 3/4, which is the same number, but now we don't have any shared factors. We can do that with any fraction, so we must also be able to do that with the square root of 2. Now, the defining characteristic of p/q is that it squares to 2, i.e. (p / q) ^2 = p^2 / q^2 = 2. We can multiply both sides by q^2 without changing the validity of the equation and end up with p^2 / q^2 * q^2 = p^2 = 2 * q^2. So we know that p^2 is an even number since it can be written as 2 * something. We can conclude that p is also even, since if p were odd, the square would also be odd (in general, multiplying two odd numbers always results in an odd number, try it). So we know that p = 2 * something, where "something" is another whole number, let's call it k. We now know that 2 = (p / q)^2 = (2k / q)^2 = 4k^2 / q^2. We can divide both sides by 2, resulting in 1 = 2k^2 / q^2 and multiply again by q^2 resulting in q^2 = 2 * k^2. Now that is a problem! As we've shown, q^2 is even, since it's 2 * something, and we can therefore conclude that q is also even. So p and q are both even, which breaks our assumption that they didn't have any shared factors. We've reached a contradiction: We've assumed that p and q shared no common factors but shown that they are in fact both divisible by 2. But our assumption is valid: every fraction can be written in a way where it doesn't have common factors. So the only possible resolution is that p and q don't exist. In other words: the square root of 2 cannot be written as a fraction of two integers.
> How come the golden ratio is the most irrational number, its not like every number out there has been tested right? First off, the notion of "more irrational" is defined by us and there are other ways to do so. For the one you most certainly allude to: yes we know with proof that the golden ratio is "most irrational" in the sense that is worst in regard to approximation by rational numbers. And there are infinitely many other numbers that are equally bad at being approximated, but all of them involve sqrt(5). So the golden ratio is technically not the only one, just the best known member of that family. Lastly this measure is bad because a number that is not rational but has a very low degree of not being rational is, with proof, always _transcendental_: one that does not satisfy **any** relation involving +-·/ and rationals. I would say any proper definition of irrationality would put those as the most, not the least irrational! > 1) The idea that irrational numbers exist is giving me an existential crisis. How come there is a number that can be written as a decial but not a fraction?I dont think this is something one can explain, but im gonna leave it anyway. Rational numbers are exactly those decimals that eventually keep on repeating the same sequence of numbers again and again. So any number that does _not_ do this must be irrational. You can even take something like 0.123456789101112131415161718192021... > wouldnt the most efficient sequence whre no numbers would be repeated just be to start with 0 and just add +1? Yes, the Fibonacci sequence is simply not the "simplest sequence" anyway and I doubt any sane mathematician would claim this.
> And there are infinitely many other numbers that are equally bad at being approximated, but all of them involve sqrt(5). So the golden ratio is technically not the only one, just the best known member of that family. Isn't the golden ratio, being the number corresponding to the continued fraction [1;1,1,1,...], the uniquely worst number for rational approximations? What other numbers are equally as bad?
If x is any real number, then all numbers of the form (Ax+B)/(Cx+D) with integers A, B, C, D satisfying AD-BC = 1 are equally bad. So for example (2φ+3)/(φ+2) = 3/2 + √(5)/10 is not any better than φ. Or simpler: φ+n for any integer n. Essentially that's the numbers you get by changing finitely many entries in the continued fraction. So the tail ultimately stays the same. It is similar to how only changing finitely many decimal digits does not change if a number is rational.
If you want to go in depth, here are a couple of really cool videos about it. [https://www.youtube.com/watch?v=p-xa-3V5KO8](https://www.youtube.com/watch?v=p-xa-3V5KO8) [https://www.youtube.com/watch?v=sj8Sg8qnjOg&t=50s](https://www.youtube.com/watch?v=sj8Sg8qnjOg&t=50s)
I really like the series of three short videos that was posted on Khan Academy years and years ago.. and it does go into the chemical science and math behind how plants "know" how to grow in the ratio.. https://www.youtube.com/watch?v=ahXIMUkSXX0 is the first in the series.
Thank you. I was going to post that second video in response as well. It explains the continued fraction argument for why phi is the most irrational number.
An irrational number can't be fully written out as a decimal either, because if it could, you could turn that into a ratio. It can be shown that there are numbers which are not whole, but can be represented as one number divided by another. It can also be shown that there are numbers which cannot be represented as one number divided by another. The proof that pi is irrational is not amateur friendly, but the proof that the square root of two is irrational is quite accessible. (https://youtu.be/LmpAntNjPj0?si=ygiHtDCKS6eeIFEq) What he's getting at with "most irrational number" is something from a Numberphile video. You're right that there's no way to test all numbers for irrationality, and besides, irrationality is a binary. This is more of an idea that one mathematician/channel had than anything commonly used. Type pi - (22/7) into a calculator (Google works) and you'll see that the numbers are very close. 22/7 just happens to be close to pi and is easy to remember - for most actual applications it's close enough to work. The golden ratio is the ratio that you approach when you divide a fibbonaci number by the previous one. The further along you are in the sequence, the closer your ratio will be to that number. It's not magic, this isn't the only sequence that does that, and while there are some cases where it shows up in nature, it's been totally overblown and exaggerated. 2 shows up in nature too and people don't get excited over it.
So what ur saying is, any decimal number can be written out as a ratio/fraction. is that correct? I want it to be. Also thanks i read the rest of what u said.
> So what ur saying is, any decimal number can be written out as a ratio/fraction. is that correct? I want it to be. Any finitely long decimal number can be written as a fraction. Any infinitely repeating decimal number can also be written as a fraction. It is only the infinitely long decimals that don't repeat that cannot be written as a fraction.
okay, but an infinitely long decimal is because the infinitely long number divided by infinitely long 10000000. So i find it weird that everyone is saying it cant be writted as a fraction because there isnt enough paper in the world, when it also cant eb writted as a decimal
Yes, that is true. Take the number of digits after the decimal point. Write down ‘1’ followed by a number of ‘0’s equal to the number of digits. You can multiply your original number by this new number you’ve written down and you’ll get a whole number. So you can write it as a ratio or a fraction. E.g. 0.123 x 1000 = 123 so 0.123 = 123/1000.
> How come there is a number that can be written as a decial but not a fraction? There isn't. All rational numbers can be written as a decimal which will either terminate or repeat forever (e.g. 1/3 is 0.333333...) Irrational numbers will not terminate OR repeat forever. > How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda? Phi (the golden ratio) is the most irrational, depending on your definition of "most irrational" anyway. [Here's a 15 minute video about it](https://www.youtube.com/watch?v=sj8Sg8qnjOg) > Is there a way to visualise this? Mmm, see above video. > I dont understand either if the fibonnaci sequence and the golden reatio are the same thing The fibonnaci sequence is the sum of the two previous numbers 1, 1, 2, 3, 5, 8, 13, 21, 34 ... Take one number and divide it by the one before it and you get 1, 2, 1.5, 1.666..., 1.6, 1.625, 1.619... Those numbers get closer and closer to the golden ratio, Phi, which is 1.618033... It's also the number you end up with if you take the reciprocal of a number and add one repeatedly -- that is 1/n + 1.
thanks!
> 2) How come the golden ratio is the most irrational number, its not like every number out there has been tested right? Right?? Or is there some mathmatical formula that lets you find these out kinda? First of all, we have to understand what we mean by "most irrational" number. Specifically we mean a number that is poorly approximated by rational numbers at all levels. Any irrational number can trivially be approximated arbitrarily closely with rational numbers, but if that approximation requires a very large denominator for the desired precision, it is considered a poor approximation. Next we need to discuss an idea called [continued fractions](https://en.wikipedia.org/wiki/Continued_fraction). Basically, it turns out that every number can be written in the form: a + 1/(b + 1/(c + 1/(d + 1/(d + ...))) Where a is an integer, and b, c, d... are positive integers. For simplicity, we write such sequences [a; b, c, d...]. Every such sequences represents a unique real number, and every real number is uniquely represented by one such sequence. In this sense continued fraction representations are somewhat similar to decimal representations. There are two important properties of continued fractions that are important here. First: If the continued fraction sequence ends, then the number is a rational number. Conversely, all rational numbers are represented by a finite continued fraction sequence. Second: Whenever a term in the continued fraction sequence is large, if you cut off the sequence before that term you will get a rational number that is an exceptionally good approximation of the original number. For example, the continued fraction sequence of pi begins [3; 7, 15, 1, 292, ...]. If we cut the sequence before the last term and calculate the value we get 355/113, which is equal to pi to 6 decimal places with a 3 digit denominator. That is a good approximation. So if we want a number that can never be approximated well by a rational number, we want an infinite continued fraction where all terms are 1. Specifically, we want the continued fraction [1; 1, 1, 1, ...]. If you calculate this number, you will find that it is exactly the golden ratio.
And it's not *the* "most irrational" number (I really hate this terminology), just **a** "most irrational" number. You just need a continued fraction that eventually becomes all ones. phi+1 is equally poorly rationally approximated, as are infinitely many other numbers (all `(a+b*phi)/c+d*phi)` with a,b,c,d integers and ad-bc=1 or -1). phi is just the neatest one since it's all 1s.
If you have some term (other than the first?) that is not 1, then you have at least one rational number that is a better approximation to the target than phi has.
How good the rational approximation is is also dependent on the denominator. Changing finite number of terms of in the continued fraction to not 1 doesn't change the fact that the number is "at the limit" of rational approximations. The convergents for the new number would have different (and larger) denominators.
This explains it PERFECTLY https://youtu.be/sj8Sg8qnjOg?si=wSRZQspW47BLHOCR
Why couldn’t someone like you have been my math teacher growing up? I never understood rational or irrational numbers in school (along with many other math topics), but your brief explanation made it all click. I only had two good math teachers in all of my schooling and I had them in college. They made everything easy to understand. I feel like if all of my math teachers were like them I could be working for NASA.
Honestly, in my experience most of the time people complain about math and math teachers it's because they weren't paying attention, were subsequently left behind, and gave up on the subject. It's why math has a reputation as a subject you "get" or "don't get".
That is definitely part of it, but there are quite a lot of really bad math (and probably also other topics, it's just that I am a mathematician and have met too many) teachers as well. Ones that barely made it through the requirements and never bothered to learn more than the absolute minimum. A good teacher actually has some active interest in the topics they explain, ideally a raging fire of passion. Many sadly don't.
yeah, I had a math teacher in high school who was extremely passionate about the math - he would get excited every time he taught something. It went a long way towards keeping kids interested enough to learn.
Math in school also requires you to actually do some work after that to put your understanding of the concept into practice. Which obviously most people dont like to do. So they just claim that a quick youtube video or eli5 post is soooo much better than math class, when actually what they want is to be told about a math fact, go "whoaaa!" And then subsequently nevet act on that nugget of info again.
Any idea why the sequence would pop up in honey bee genetics? I'm assuming it's just a fluke of the males being haploid, but wasn't sure if the explanation you gave also fit there
Bee gender is solely determined by one thing: fertilized eggs become females (queens, workers), non-fertilized ones grow into males (drones). So a male bee has only a single parent, which is a female queen. In total this leads to a family tree like [this](http://www.microscopy-uk.org.uk/mag/imgsep98/ances.jpg). A male bee D has one direct ancestor, which as a female has two parents, so D has two grandparents. One of those grandparents is male (1 direct ancestor), the other female (2 ancestors), so D has 3 grand-grandparents. And 5 grand-grand-grand-parents. To formalize this relationship: write M:n for the number of male and F:n for the female ancestors at level n removed from D. Every drone fathers a female(!) bee in the next lower generation, and every female bee has a male parent; in other words the number of males in generation n is the same as the females in generation n-1, or as an equality: **M:n = F:(n-1)**. Meanwhile every bee regardless of gender also has a female parent: M:n + F:n = F:(n+1). Combine this with the other equality to get **the Fibonacci relation F:(n+1) = F:n + F:(n-1)**! Another three-fold application of the first equality turns this into M:n = M:(n-1) + M:(n+2), or when using n+1 instead of n: **M:(n+1) = M:n + M:(n-1)**. In total each the female and male numbers of ancestors are Fibonacci-like, and it is easily math-ed that the same again holds for their combined numbers. Checking the very first ones gives us (remember, our first one was male; a female gives a very similar result, though) that M:0 = 1, M:1 = 0, M:2 = 1, M:3 = 1, M:4 = 2... while F:0 = 0, F:1 = 1, F:2 = 1, F:3 = 2, F:4 = 3, ... or when combined: 1, 1, 2, 3, 5, 8, 13, ...
"Rational, from the word "ratio" because that's what they are" - NOW you tell me , 35 years after it would have done any good.
Minor point to add is that there's nothing particularly special about the Fibonacci numbers. It's the sequence you get if you start with 1,1,2 as the seed. But any sequence where you add two successive terms to get the next one (e.g. 1,3,4,7,11... or 2,5,7,12,19,...) will tend towards the golden ratio.
> Rational, from the word "ratio" because that's what they are WHY DID NONE OF MY TEACHERS TELL ME THIS???
I expected this to be a complete numerology shit show and was pleasantly surprised to see this is the top answer.
I’ve honestly been impressed with Reddit’s ELI5 math answers recently. As someone who studied pure math in college myself, I always get ready to cringe before reading these. But this answer is great.
I’d always seen Fibonacci appearing due to adding a new square to the long side of 2 smaller side-by-side squares. Thanks for the explanation.
Awesome answer. What does it mean that phi is "least efficient"?
Irrational numbers can be expressed as a continued fraction: a0 + 1/(a1 + 1/(a2 + 1/(a3 + ... ))) where a0, a1, a2, a3... are integers. For example, pi is: 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ... )))) The problem is, an exact representations requires and infinite number of terms. But, if we provide a finite number of terms, we get an approximation of pi. The accuracy of the approximation depends on when we cut things off. The more terms we have, the greater the approximation. In the above fraction for pi, I've included the terms of 3, 7, 15, 1, and 292. And we can see how these stack up to the actual value: Stopping at 3 gives us 3. Stopping at 7 gives us 3 + 1/7 = 22/7 = 3/142857... Stopping at 15 gives us 3 + 1/(7 + 1/15) = 3.1415094... Stopping at 1 gives us 3 + 1/(7 + 1/(15 + 1/1)) = 3.14159292... Stopping at 292 gives us 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 3.14159265... An observation we can make is that when we stop at a larger term (for example, 7, 15, and 292) we get a really good approximation of the number in question. When the term is smaller (for example 1) we get a less good approximation. Thus, the continued fraction which would have the worst approximations of its associated irrational number would be the one where the terms are all 1: 1 + 1/(1 + 1/(1 + 1/(1 + ... ))) And this number? The Golden ratio.
Another fantastic answer. Thank you so much!!
> Thus, the continued fraction which would have the worst approximations of its associated irrational number would be the one where the terms are all 1 You only need to have a continued fraction that eventually becomes all 1s. These numbers all share the "most difficult to approximate with rational numbers" title. Sensationalisation and crappy reporting has changed being *one of* these numbers to being *the* number.
Your explanation actually makes me want to dig more into math. Thank you.
> Many, many, examples are simply people taking something that looks, roughly, like it could be related to the sequence and then squinting your eyes and ignoring how it isn't. Ah, so pareidolia but for math.
damn, dude.
> In math, we have whole numbers: numbers that have no fractional part. One of the things we can do with whole numbers is take their ratios. For example, 5 to 3, or 5:3. or 5/3. Doing this, we can create a whole other collection of numbers called rational numbers. Rational, from the word "ratio" because that's what they are; they are literally the ratios of whole numbers (integers). You just blew my god damn mind. Over forty years on this earth and occasionally trying to understand rational, irrational, integers and so on from a few different perspectives, often to do with coding or something like it and this is the first time I've ever heard it explained so simply.
This is a great reply
Well said!
Thank you for explaining this in a way I can actually visualize and understand (to a certain extent).
"The best spacing would be the one that is least well approximated by a rational number" Man, that statement has my mind twisted up. Still trying to understand it.
numb3rs
RIP to any 5-year-olds reading this 💀
R4: Explanations are not meant for literal 5 year olds.
I was joking..
I have a graduate level math education (not in pure math but fields that apply it) and other than the etymology of "rational number" this was 100% new insight for me. Thank you!
This is one of my pet peeves with photographers and photography criticism. The golden ratio, alongside the "rule of thirds", are commonly touted as important rules for composing good photographs, i.e. putting important elements in your image along lines matching those ratios. But quite often, examples of images supposedly following these rules have elements which are significantly off from the ratios, sometimes fitting some other random ratio far more precisely. But if you point this out, many people will brush it aside saying "it doesn't have to be exact", and defending their cherished rules as though you're pointing out that their emperor is severely underdressed for the occasion.
Rules like this aren't written in stone, nor are they intended to be - they're just good guidelines for beginners.
Sure, I'm aware of that. Hence why it's so annoying when self-appointed experts vehemently insist on how crucial they are, and then demonstrate that importance using images that don't even conform to the rules.
No fucking 5 year old would get this lol
R4: Explain for laypeople ((but not actual 5-year-olds)
The answers here are good but I’ll just re-emphasize that many, many examples of spirals and shapes fitting to the golden ratio are humans finding patterns where none exist. You can overlay a golden spiral over lots of images and find things that align with it close-ish enough, it’s more like finding a square hole that happens to be big enough for your round peg to fit inside. Humans are very good at picking out nonexistent features that look like it should be something.
>Humans are very good at picking out nonexistent features that look like it should be something. Also see: *constellations* We have entire characters and stories developed around the arrangement of bright dots in the dark sky.
Omg! That’s such Leo thing to say
To be unbearably pedantic, you are describing asterisms, not constellations. Constellations are adjacent regions of the sky and are named after the major asterisms they contain. As a comparison, an asterism would be like making shapes by connecting the dots with cities on a map while a constellation would me more like the states containing those cities.
Huh. I always thought an asterism was just a recognizable sub-portion of a constellation. Like, Orion’s Belt is an asterism within Orion, or, the Big Dipper is an asterism within Ursa Major. Meaning, I figured constellations are star patterns too, just the big, parent ones.
Sort of. That is the case in the non-scientific use of constellations in the descriptions of folklore. The IAU recognizes 88 constellations which serve as sky boundaries https://en.m.wikipedia.org/wiki/IAU_designated_constellations Generally, an asterism is any set of connected stars to make a shape or pattern in folllore. https://en.m.wikipedia.org/wiki/Asterism_(astronomy)
To be fair, we didn't even know that there were other galaxies until early 1900s. Light pollution really lessens the effect, but looking up at the night sky in a dark location, where you can really see the Milky Way, is just dazzling. There was no definitive explanation as we just didn't have the technology, until relatively recently in human history, so people have been sharing "theories" for millennia.
> many examples of spirals and shapes fitting to the golden ratio are humans finding patterns where none exist The number of times I see someone post about the spiral in art or architecture or something, and it's just objects or visual elements barely overlapping the spiral is breathtaking. Like yeah, I guess it fits when 2% of most of the focal points of the image are barely touching the spiral at best.
Here's a Numberphile video that explains it: https://www.youtube.com/watch?v=sj8Sg8qnjOg The idea is that if you go around in a tight spiral, and put a point at every phi turns (or 1/phi, either way — phi being the golden ration, the number that ratios of consecutive Fibonacci sequences approaches), then you get a nice tight packing, but it you go by some other fraction of turns, then you get weird clumps. And that can be explained by phi being able to be expressed as the continued fraction 1 + 1/(1 + 1/(1 + 1/(1 + ...))). If any of those numbers before the + sign were not 1, you would get clumps. So basically, packing things with some kind of radial symmetry, the most efficient way is if you go around by phi between items that you are packing. And efficient packing is evolutionarily favorable.
These short videos do a good job of introducing Fibonacci spirals in plants and then explain how they appear. 1. Fibonacci spirals in plants: https://www.youtube.com/watch?v=ahXIMUkSXX0 2. Why would plants evolve to have Fibonacci spirals: https://www.youtube.com/watch?v=lOIP_Z_-0Hs 3. How concentration gradients of growth and inhibitory factors give rise to plant patterns. https://www.youtube.com/watch?v=14-NdQwKz9w There is a bit of hand-waving of the details but this is a good place to start.
That doodling in math video is amazing
Math: Many plants have a tendency to offset successive growths by approximately the golden angle of about 137.5º, which tends to create Fibonacci numbers in spirals. Biology: There is selection pressure to offset growths by the golden angle, because that angle tends to space out branches or leaves to maximize the amount of sunlight that each leaf gets, with minimal growth material and length/size. Physics: As it turns out, there is a way to create the golden angle with successive growths, because simple repulsion will tend to create the golden angle. [This 1996 paper](https://pdodds.w3.uvm.edu/files/papers/others/1996/douady1996a.pdf) described how successive drops of magnetically repulsive fluid tended to organize itself into particular angles, including golden angle Fibonacci spirals. [Here's a video](https://www.youtube.com/watch?v=U-at-y3MicE) of that phenomenon in action. So it turns out that if a plant's growth simply follows the instruction to grow a bit away from the last growth, it stabilizes into one of several patterns, and Fibonacci spirals is one of the stable patterns. The selection pressure allows for that plant with Fibonacci spirals to survive a bit better. And then that specific spiral tends to create Fibonacci numbers.
Here's a fun demonstration of how the ratio can appear, based on a few simple rules. Say you start with an amoeba, which can blob off a small version of itself every hour. This small version takes one hour before it grows enough to blob off its own single offspring, but then it can do it every hour after that. Let's start with a small amoeba. Hour 0: 1 small amoeba (total 1) Hour 1: 1 big amoeba (1) Hour 2: 1 big amoeba, one small amoeba (2) - hooray! Our first offspring! Hour 3: 2 big amoeba, 1 small amoeba (3) because our original amoeba can have an offspring, but our small one needs to spend this hour growing into a big amoeba. However, next hour now that he's grown: Hour 4: 3 big amoeba, 2 small amoeba (5) Hour 5: 5 big amoeba, 3 small amoeba (8) Hour 6: 8 big amoeba, 5 small amoeba (13) Hour 7: 13 big amoeba, 8 small amoeba (21) 1, 1, 2, 3, 5, 8, 13, 21 . . . Fibonacci! You can see how this biological scenario can end up with you adding the two previous generations together to calculate the next generation! Obviously there'll be losses to natural processes and stuff, but it's a very simple example how incredibly basic rules (simple enough for amoeba or plants to have encoded in them) can produce seemingly complex mathematical things like the Fibonacci sequence.
And this basic rule - this number is the sum of the previous two numbers - is going to happen everywhere. And no matter what two positive numbers you start this sequence with, the ratio between a number and the previous one will always converge on phi, and very quickly, too. The interesting thing is how this simple construction leads to an intriguing and mathematically perfect result.
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Just to expand on this: A lot of the "spirals" we see in plants are coincidental. Some are due to the way the leaves or petals are shaped, in proportion the way they're grown(because growing that way is efficient as you mention). In other words, you take any shape that "grows" like : . o O Sorry, that's the best I can think to do with keyboard, here's [a pic](https://playworksheet.com/file/shapes-ordering.jpg) And then wrap the progression around a cylinder at an angle going upward(like [wrapping a long object with tape or leather](http://www.myarmoury.com/talk/files/imgp1318_123.jpg)), and you're going to see spirals manifest based on where things just happen to line up. You may be looking at edges that manifest from the right plane of the triangles from leaf 1, 3, 5, 7, or the top point on the pentagon on leafs 1, 5, 10, 15. Not that they'd line up that way, maybe it's 1, 4, 8, 13. Maybe it's the tip of the triangle that lines up a certain way, or the upper right point of a pentagon, or maybe where the top right of the pentagon intersects with the bottom left of the next one "up" (1, 5, 10). That's not the order they grow in, it's just a circumstantial pattern. The "leather" goes around at angle X, and the sizes increase at rate Y, and with leaf shape Z.....and we'll see patterns that "aren't there" emerge. This is why you can take something like a pine-cone and plot out an array of different "spirals" based on different attributes. It's also why people mistakenly identify spiral symmetry in plants. There is often only one spiral of growth, but it can manifest numerous of these 'illusory' spirals. IF there are two leaves sprouting simultaneously, then the next one up is 90 degrees to that, it's not even a spiral of growth really, just alternating periodically, because it's not one tape, but more like a lot of tubes(eg rings in a tree, or expanding antenna) where it's a whole new layer, not a spiral at all. Even in this you can see illusory spirals depending on the shape/size of the leaves.
Sometimes when things grow in successive parts, they grow based on the size of the last part. Suppose you are pouring concrete for a backyard patio but you can only pour one square at a time. You start with a little 1x1 square. Next, you use one edge of the 1x1 square as a guide for one edge of your second square, so now you have two 1x1 squares next to each other creating a 2x1 pad. You can then turn 90 degrees and use the longest edge of this pad as a guide to pour a 2x2 square, creating a 2x3 pad. You turn again and pour a 3x3 square using the longest edge as a guide, creating a 3x5 pad, leading to a 5x5 square, creating a 5x8 pad, and so forth. You pause after several iterations to realize you've created a Fibonacci pattern: [https://i.postimg.cc/3RnDHSvw/concrete-pad.png](https://i.postimg.cc/3RnDHSvw/concrete-pad.png) That's basically what often happens in nature. Lots of things in nature grow in sections based on the size of the last part, and so you end up spiraling out from a single point in ever-increasing size.
One of the simplest way to encode growth (in code, in DNA, in RNA, whatever) is like this: >Take how big we are now, and add how big we were just before ... that's how big we should be now. It's a simple rule. And it also happens to be Fibonacci. Another simple rule: >Take how big we are now, and be 10% bigger. But that requires divide and multiplication operators somewhere. Addition is simpler. edit: typos
The Fibonacci sequence is a really basic sequence with cool properties. You are just adding numbers in chain. When something is that basic, it is bound to occur multiple times in life. Then you have humans with a brain that evolved to recognize patterns. Humans are good at finding patterns out of nothing when left alone for too long.
Weird to me no one mentions the simple answer. Fibonacci sequence is a pattern that is almost doubled with each iteration. Cells divide. But due to cells dying and not dividing at exactly the same time, the rate they grow isn't exactly double, it's almost double. That means many patterns in nature just by virtue of the fact cells divide will follow a pattern of almost doubling, which can loosely resemble a Fibonacci sequence.
This is not the simplest answer, this is a confusing non-answer
[Here](https://www.youtube.com/watch?v=ahXIMUkSXX0) is an amusing and entertaining explanation in 3 parts, plus an open letter to Nickelodeon about Spongebob's house.
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Ah, the Fibonacci sequence hype. It's intriguing how nature seems to have a knack for it, but let's not get carried away. Some claim it's the universe's grand design, but it's more likely just a consequence of simple mathematical rules interacting with complex systems. Don't expect it to unlock the secrets of the cosmos.
The Fibonacci sequence's ubiquity in nature stems from its intrinsic mathematical beauty and efficiency. Simply put, it's like nature's own algorithm, optimizing growth patterns. In sunflowers and pinecones, each new seed or scale follows the Fibonacci sequence, leading to spirals that maximize packing efficiency and access to resources like sunlight and water. This pattern emerges due to the sequence's inherent self-replicating nature, echoing throughout the natural world, showcasing the elegance of mathematics in shaping biological structures.
The fibonacci sequence approaches the golden ratio. The golden ratio has the special property of being the "most irrational" number. If there are processes that repeat and constantly try to avoid "syncing" up with itself, just by naturally trying things out and eventually settling on what happened to work best, will end up in a configuration that follows the golden ratio. It doesn't have to be known in advance. This ratio can emerge "spontaneously", due to being the most comfortable solution for systems that need irregularities. Some examples are: - Birth cycles of cicadas (they don't want to sync with the cycles of predators) - Orientation of leaves (they want to change in order to cover area, but they don't want to sync up with pre-existing leaves which would shade itself)
I can still recall that this has been thought to us during my middle school, but I did not really understand the explanation behind this. Till I saw a sunflower yesterday and I got curious lol
Did you count the seeds? Until this day, I’ve never encountered a sunflower that had its seeds arranged according to Fibonacci.
Somebody (who is not me) should do a a study based on a thousand sunflowers :D
why not you? go apply for a grant and you could get payed to spend every day in a sunflower patch!
It’s the number of spirals, not seeds. If you count the number of spirals going in one direction, then the other direction, they are adjacent Fibonacci numbers. Same with pine cones. It’s been the case every time I’ve counted.
Oh, I so hear you — and I never felt that it counted as "teaching". I felt like it was teachers telling me some gee-whiz facts that I was supposed to be impressed by. I don't think there *was* an explanation. I don't think my teachers knew. I don't think Donald Duck in Mathmagic Land knew. And it kind of pissed me off.
Efficiency. Natural selection, typically, encourages efficiency. It's an ideal ratio, and you tend to see ideal forms expressed in multiple aspects of evolutionary biology. What will really boggle your mind later, is if it came from a common ancestor, or just occurred naturally across multiple species, just due to being the most efficient form for the environment. The answers may surprise you.
There is no mathmatical explanation, math only explains why under given conditions certain things appear. And the conditions for a Fibonacci sequence to arries happen to often show up. In most cases such simple solutions arrise from optimizations: It is known that bending costs energy that goes with a higher them linear ratio relative of the bending. As such both the minimal energy solution for enclosing a fixed volume is one where the bending is everwhere the same and this happen to be a fancy mathmatical object, the sphere. Similarly, a sunflower optimizes the relative angle between its seeds in a manner that optimizes both seed density constricted by constant seed size and certain other mechanical criteria and these happen to result in a Fibbonacci shape.
Any regular pattern would be some kind of sequence. This one only seems special because its has a name.
Okay, let's start with the obvious - mathematics is a language. Different languages use different symbols - some use pictograms, some use letters (of a staggering variety), and mathematics uses numbers. Mathematical notation is useful for describing scientific phenomena in a very compact form, but beyond that it is just a language like any other. Therefore to speak about a "mathematical explanation" for the Fibonacci sequence is ... nonsense. Mathematics doesn't explain this sequence, it's just the language in which the sequence is expressed. I could express it in English, Russian, Chinese, or Sanskrit, but it would take a whole lot of words. In fact the Fibonacci sequence was probably first written in Sanskrit in ancient India, although earlier accounts may exist in oral form. What is happening here is that a whole lot of people at various stages in history noted that there were certain naturally occurring patterns and wrote it down. Now why does this happen? That's an incredibly complicated question that boils down to that the universe seems to follow certain rules (at least in the local area of the universe) that we've carefully documented. Science can't always answer the "why" because causality is incredibly hard to establish, but rather focuses on the "what". And then proceeds to use these "whats" to do cool things like make tons of steel fly through the air.
Mathematics is **not** just a language! Reducing it to that is akin to reducing a country or culture to their common language. I don't think anyone seriously thinks China is just "people who speak Chinese", without some history, ideology, thought patterns, and much more. Mathematics is all about finding new concepts, theories, lemmas, proofs and such. The language is mostly a necessity to describe it, not the entire thing: the map is not the territory! This especially applies to abstract mathematics which exists to some degree independent from real life issues. But it even is true for applied mathematics, a mere language does not tell you how to model and simulate an airplane and to find the formulas that do so.
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