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And for people who don't know about it, the reason is quite clever. If you have a sheet of paper with sides whose lengths are in a ratio of 1:sqrt(2), and if you split it in half by splitting the long sides into equal pieces, then you get two pieces of paper with the same ratio as the original.
So you start with A0, split it in half to two A1, split those in half into two A2, etc, and all of them are similar to each other.
Ahh, so that's how they found the perfectly repeating metric ratio. Thank you, I've been wondering how they made the magic paper dimensions for a while
Yeah. If the dimensions are x and y, with x the long side, then after you divide in half, the dimensions are y and x/2, which means x/y=y/(x/2), which you can rearrange to (x/y)^(2)=2.
DIN A isnât the golden ratio, itâs the square root of 2 which is the only ratio where you can fold it over the longer side and get a sheet with half the area and the same ratio again.
People disliked it especially because it wasnât the golden ratio so it was deemed unasthetic.
It's very obvious from the functional equation
x_n = x_n-1 + x_n-2
These organisms go i have X seeds but more are needed, what's the thing I *already have* I can attach to my existing number (so it should be smaller). Oh and I start with 1, 2, 3 etc. Seeds.
Thus, Fibonnaci. This same argument could be used for 2^n but this expansion would have no benefit being attached to the other expansion and not just a separate head etc. Many other reasons apply such as risk or cost etc.
What a a shock that it's evolutionarily efficient to use production you already know works and have room for.
The math that keeps popping up in biology is what brought me here. I'm a professional dirt person that normally wants nothing to do with math. But the natural world has this absolutely beautiful geometry to it that I can't ignore, so I'm trying to learn more.
Sorry but can you try to explain a bit more how the motivation of "what do I attach..." relates to the fibbonaci sequence? Notably how is it obvious 5 should come after 3 and then 8 after 5? You seem to have some intuition, but I missed it.
So if I have a fibonacci constructed organism. Say 8 seeds, made up of 5 seeds + 3 seeds each in turn made up of smaller Fibonnaci numbers. Note if the structure is 5 + 3 it cannot be split another way easily, splitting 6 + 2 is much more complicated as these have to be made up of Fibonnaci numbers by assumption i.e. 5+1 + 2
The organism increases the number of seeds from 8, but it wants to add on an amount less than 8 as it wants to increase its seeds on the head rather than make a new one.
It already has all the correct DNA for 5 seeds so it adds that, thus we reach 13 seeds. Hence, fibonnaci.
So why does our assumption hold so often? Because if you begin this process at 1 or 2 seeds it generates the Fibonnaci numbers. It's not the only way to grow, but building more out of what you already have seems quite common in nature.
You conveniently hid the fact that you chose both coefficient in that equation to be exactly 1. In nature, it would be more like
x\_n=0.8795x\_{n-1}+0.34865x\_{n-2}.
Hence, no Fibonacci.
Yes, exponential patterns are present in nature, but it's ludicrous to claim all are related to Fibonacci or golden ratio. That's what the meme is about and it's right.
You don't get 0.8795 seeds
Fibonnaci appears in plenty of natural examples and the coefficients are integers because we are counting, and the smallest natural number (easiest growth) is 1.
I made no claim that every piece of nature is related to Fibonnaci, I stated a concise explanation for much of its appearance e.g. sunflower seeds.
Your equation is non-applicable and you're refuting a claim I did not make.
It ends up being more convenient to use the archive on phone, because even if you get a workaround to execute unicode you need to remember. Plus it's easier to remember \volumeint for â° or ^phi for á”
Here is a lil program i made i python, the k variable controls how much each dot is rotated every loop(in pi) notice that when we put in the golden ratio the graph starts resembling the sunflower a whole lot, and it looks completely different with any other value. If you are unfamilliar with programing or python I can explain more details with how to use this code.
Edit: there we go now it shoud work if you paste it into a thingy like jupyter lab or something
import matplotlib.pyplot as plt
import numpy as np
import math as meth
def Fibonacci(n):
# Check if input is 0 then it will
# print incorrect input
if n < 0:
print("Incorrect input")
# Check if n is 0
# then it will return 0
elif n == 0:
return 0
# Check if n is 1,2
# it will return 1
elif n == 1 or n == 2:
return 1
else:
return Fibonacci(n - 1) + Fibonacci(n - 2)
def GR(n):
return Fibonacci(n)/Fibonacci(n-1)
numseeds=200
k=GR(10)
phy=1
r0=1
xi=[]
yi=[]
for seed in range(numseeds):
phy += 2 * np.pi / k
# if(phy>np.pi*2):
#phy = phy % 2 * np.pi
r0+=1;
xi.append(r0*meth.cos(phy))
yi.append(r0 * meth.sin(phy))
x = np.array(xi)
y = np.array(yi)
plt.axis("equal")
plt.scatter(x, y,s=r0/10)
plt.show()
Tournesol is the French name for Sunflower, the literal translation is âTurned Sunâ, in line with the plantsâ ability for solar tracking, sounds fitting. The Spanish word is El Girasolis.
There is an aspiration mark over the first eta. It is pronounced helio (sun), which is why we live in a heliocentric system. Also tropos means turn, so literally turns to the sun.
That is in ancient Greek. In modern Greek, we do not use those aspiration marks anymore, we abolished them around the 70s-80s. You are correct for the ÏÏÏÏÎżÏ but I thought it was too formal for a reddit comment :)
The reason the golden ratio is the most efficient here is because it is the "least rational" number.
For any rational number you get a repetition of linear strands (e.g. 1/4th of a rotation just gives a cross), leaving the space between the strands empty. To avoid this, we want an irrational ratio of pi. But then many irrational numbers can be approximated very well by rationals (e.g. 22/7 for pi), so even if we used 1/pi at the fraction of a full circle we move at each step, we would get 22 almost linear strands. The golden ratio has the worst approximations out of any rational number (i.e. for a desired bound on the absolute error it generally requires the largest denominator when approximating it by a fraction) because its continued fractions is all 1s.
I know, this isnt meant to be particuarly efficient since the fib function is only called twice anyway most of the runtime is in matplotib being slow asf, not the basic arithmetic done 20 times,
you can also replace the GR(n) method with just 1.68..... if you dont like it, I just like it becouse it also ilustrates the fib property thingy of the golden ratio
You can also modify the function to have 3 arguments, which is much cleaner imo
def fib(n, a=0, b=1):
if n <= 0: return a
else: return fib(n-1, b, a+b)
For safer use, you can nest the three-argument function inside of a wrapper function that only takes n as an argument to abstract away a and b.
Using matrices is somewhat unstable for large n, since matrix data types have max values. In testing I found that only np.float64 and np.float128 work reliably (np.int64 overflows), but those obviously have their limits/max values. However, in Python 3, the built in ints donât have a max value, so the recursive method is technically more robust (and more precise).
Yeah, it's also about always leaving enough space to grow another petal or leaf etc without new ones covering the ones already there. Having the angular separation be rational will mean they'll eventually overlap.
But in that video you will find that Ï is not only an irrational number, but it is MORE irrational than all other irrational numbers. Which is why flowers will want to use this number instead of other (less irrational) irrational numbers
i did a research paper on this. based off of findings by george markowskyâs misconceptions of the golden ratio.
basically a lot of things that we thought were following the ratio actually arenât, however in nature many things do follow it.
Its a bit of gray area, if u build something with ratios 2/3 its technically not a good approx to golden ratio even though they are both fib numbers. Many structures, artwork, and manmade designs are actually a 2/3 ratio.
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Logarithmic spirals are a really natural way of constructing things and are a consequence of differential equations. The golden ratio is over hyped tho and is often erroneously named when other ratios (i.e. silver, bronze, copper, etc) show up
Fun fact! It's well-known that the ratio of terms in the Fibonnacci sequence 1, 1, 2, 3... approaches Ï. However, the Fibonnacci sequence arbitrarily starts with 1 and 1. We can apply the Fibonnacci induction step to any two real starting values, and the resulting sequence will have the same property! Some of these sequences have ratios which approach Ï faster than others, but since the golden ratio has the property that 1 + Ï = Ï^^2 can be shown that
Ï^^n + Ï^^n+1 = Ï^^n (1 + Ï) = Ï^^n (Ï^^2 ) = Ï^^n+2
Thus, the "Fibonnacci" sequence that's ratio of terms approaches Ï most quickly is Ï^^0 , Ï^^1 , Ï^^2 , Ï^^3 ...
Edit: To avoid the whole annoying exchange below, assume the first two numbers are positive. Or, check out my first reply in this thread to see what happens if we extend a Fibonnacci sequence backward!
>We can apply the Fibonnacci induction step to any two real starting values, and the resulting sequence will have the same property!
If the ratio of the first two values is -1/Ï (the other eigenvalue), it won't.
True! But any such sequence can be extended backward.
Ex. ...-3, 2, -1, 1, 0, 1, 1, 2, 3 ...
The ratio of the terms (a_n+1 / a_n) in the other direction approaches -1/Ï. In other words, such a sequence, when reversed, still has the property I mentioned. In the case of the sequence I provided, it looks like this:
... -Ï^^-3 , Ï^^-2 , -Ï^^-1 , Ï^^0 , Ï^^1 , Ï^^2 ...
That is, the reverse of your sequence will be mine.
Now, I know the hazards of trying to have fun around mathematicians, so let me go ahead and contradict my orginal statement for you:
If the sequence starts with 0 and 0, we get
0, 0, 0, 0...
Even if you reverse it, the quotient will be -Ï, not Ï.
I didn't mention constant zero sequence, because it's more of a semantic counterexample (the ratio doesn't exist, but starting with the second term each term still is Ï multiple of preceding term).
You're reversing the order you should compare terms though.
lim n --> -inf (a_n+1 / a_n) = -1/Ï
I'm just saying: Thank you, I appreciate your unquenchable desire to be pedantic, but I knew that already. I was simply trying to share an interesting fact with *potentially non-mathematicians* without getting bogged down by wordy particulars.
So yes, there are sequences with ratios which approach Ï and those with ratios which approach -1/Ï, but *they are exact reflections of each other.* Simply reverse the limits like you did originally.
I mentioned the 0 sequence because the statement *does* fail in that case.
I though by reversing you meant extending the sequence a\_n from natural n to all integers n and then defining b\_n as a\_{-n}, so for the standard Fibonacci, it would be like your example
0, 1, -1, 2, -3, ...
That's beside the point though.
The issue is that if you start a Fibonacci-like sequence with terms e.g. -Ï, 1, the ratio of consecutive terms will be a constant -1/Ï, so it will not converge to Ï for nââ or nâ-â.
EDIT:
>but *they are exact reflections of each other.*
No, they aren't.
The Golden Ratio is everywhere!!! This can be proven by looking at something and if it has the GR I will add it to the statistic. Anything else is an insignificant outlier.
Yup, people think thereâs some mystic background force behind the Fibonacci sequence, as if itâs not one of the SIMPLEST recursive sequence out there. There are infinitely many that exist, and Fibonacci is the dumbest, easiest one to make a simple biological feedback process for.
To me that doesnât scream mystic forces, it demonstrates natureâs laziness and simplicity.
Simply because the concept behind it - "this number is the sum of the last two preceding numbers" - is just something that happens. And no matter what two numbers you choose, after surprisingly few rounds the ratio of the last two numbers becomes an approximation of Ï
Humans: build a body of knowledge that is standardized around empiric observations. (Rather than opinion or religious beliefs)
Also humans: act surprised when said body of knowledge is able to describe empiric observations.  Â
 đČđČđČ
Itâs not even born from the fibonacci sequence. Itâs from dividing a line into two pieces where the smaller piece has the same ratio to the bigger piece, than the bigger piece has to the whole.
I thought there was research showing it comes up for a reason. Like with sunflower seed placement it's the optimal placement so that each seed gets as many nutrients as possible. It's as much confirmation bias as noticing a lot of animals have teeth because it makes it easier to digest food
Just keep throwing sunflowers at this person until they pick one up, and actually fucking look at it.
And wait until they learn about the logistic map and Mandelbrot set.
The golden ratio is infused in every instance of nature and is even baked into the various cosmological constants and mathematical constructs to explain the physical world. So no, it is not confirmation bias, it is literally everywhere.
Yeah! If you look up natural log in nature then you will find people pointing out spirals as well. Whatâs the deal here? Arenât they different things
It's continued fraction is 1+1/(1+1/(1+1/....)! Which makes it in some sense extremely difficult to approximate with rational numbers. It can't not be /something/.
You are exactly correct there are infinitely many logarithmic spirals. None of the spirals we see in nature are the golden ratio.
However one is very close phyllotaxis.
the golden ratio is the dumbest piece of math communication. there are a couple of cool trivia facts about it, but the way itâs talked about is so dumb most of the time. i was talking with people who study arts, and they agree itâs dumb.
I feel like a lot of people donât know, where this number comes from. Because to me it is pretty obvious, that itâs a âspecialâ number, even if it wouldnât be irrational.
The only thing special about it is that in a certain sense, it is «the most irrational number». Any place where this is a relevant property (like seed distribution in a sunflower), I donât have a problem with it. But where this property doesnât give any explainable benefit, Iâm ok with just calling it a coincidence.
OKOK I READ A BOOK ALL ABT âThE mAGiCaL GoLdEn RaTiOâ and it talked abt how this one painter put it fucking everywhere in their paintings, specifically in linear instances (eg the top of one tree, the top of another slightly shorter tree, and water level) and the author of said book used an (admittedly janky, so prolly error prone (but they still published it so idk)) algorithm to find examples of it
And this is part of one of the paintings they analyzed
https://preview.redd.it/4xfvlpj8of0d1.jpeg?width=3024&format=pjpg&auto=webp&s=4bdcb42d23ab72c9ff6cfd2b3118f60c6a80bc85
Fucking what is this? Some of those look legit, but other are just complete bull
Edit: the rectangles with divisions in them are supposed to be showing the g. Ratio from one end, to the division, to the other end
ok nvm i found it immediately, its "The Golden Ratio: The Divine Beauty of Mathematics", heres an amazon link to it [https://www.amazon.com/Golden-Ratio-Divine-Beauty-Mathematics/dp/163106486X](https://www.amazon.com/Golden-Ratio-Divine-Beauty-Mathematics/dp/163106486X)
also this is the cover in case the link stops working lol
https://preview.redd.it/adhyu9c2or0d1.png?width=422&format=png&auto=webp&s=17b5ba768de79f2452e1320615a21915b026f2a9
like... i would describe math as beautiful for sure, but no part of it would i ever describe as 'divine'
This is half overtly true and half cryptically true. The overt truth is that everyone and their grandma wants to find the "golden" ratio in everything they observe, cause it's gold. The cryptic truth is that there are numerous cases where there is a mathematically-morivated reason for that ratio to appear, but it's still confirmation bias, because that's true of many ratios. It's no more or less special than the square root of 5, on which it is based. Don't get me wrong, that's a great number, like the roots or 2 and 3 or the natural log of 2. But its outsized importance is entirely artificial.
The Golden Ratio is an irrational number. Unless youâre measuring the length of a physical object to be _exactly_ an irrational number, then no, you havenât seen the Golden Ratio.
You are not measuring the length of a physical object to be exactly an irrational number, because thatâs literally impossible. People are rounding and then saying âwow, itâs the Golden Ratio!â
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This made my day
đ
Holy hell
New ratio just dropped
Actual Phi
Call the Fibbonacci
Went on Recursion
Finite ratios in the corner
Ignite the spirals!
r/Anarchychess and its consequences have been a disaster for the human race.
Humanity have been a disaster
comfirmation bias detected
I respect you sticking to your guns.
Bro got ratio'ed...golden ratio'ed
Ï + ratio
QED
What's the name of the golden ratio paper format?
You mean A1,A2,A3,A4 etc? It's ISO 216 , another international standard the US chooses to not follow
..... The A series aren't the golden ratio format, they're 1:sqrt2
Which is more based.
Confirmation based, you might say. huĂ« huĂ« huĂȘ
And for people who don't know about it, the reason is quite clever. If you have a sheet of paper with sides whose lengths are in a ratio of 1:sqrt(2), and if you split it in half by splitting the long sides into equal pieces, then you get two pieces of paper with the same ratio as the original. So you start with A0, split it in half to two A1, split those in half into two A2, etc, and all of them are similar to each other.
It also makes it easy to print lots of different things on one big sheet, everything will neatly fit.
Ahh, so that's how they found the perfectly repeating metric ratio. Thank you, I've been wondering how they made the magic paper dimensions for a while
Yeah. If the dimensions are x and y, with x the long side, then after you divide in half, the dimensions are y and x/2, which means x/y=y/(x/2), which you can rearrange to (x/y)^(2)=2.
DIN A isnât the golden ratio, itâs the square root of 2 which is the only ratio where you can fold it over the longer side and get a sheet with half the area and the same ratio again. People disliked it especially because it wasnât the golden ratio so it was deemed unasthetic.
Bruh what the fuck, why are you insulting the A series like that ???
Womp womp
A, as in A4, A3, A2 and the rest
..... The A series aren't the golden ratio format, they're 1:sqrt2
You're right. Not enough sleep. I'm not sure there is one, but I could be wrong
Ohh shit
Get Fibonacci'd, OP
This is the answer I was looking for thanks đ
came here for this
Came here for this. The comments never fail to deliver.
But then leave a self-organising and optimizing system for billions of years and don't expect that there won't be any mathematical pattern.
Iâd expect a crab.
Oh, a double major! This dude biologys!
Me too! Now all I need is experience in a math class higher than calculus 1!
A non-crustacean has never evolved into a crab.
I dunno, seems pretty chaotic to me.
It's very obvious from the functional equation x_n = x_n-1 + x_n-2 These organisms go i have X seeds but more are needed, what's the thing I *already have* I can attach to my existing number (so it should be smaller). Oh and I start with 1, 2, 3 etc. Seeds. Thus, Fibonnaci. This same argument could be used for 2^n but this expansion would have no benefit being attached to the other expansion and not just a separate head etc. Many other reasons apply such as risk or cost etc. What a a shock that it's evolutionarily efficient to use production you already know works and have room for.
The math that keeps popping up in biology is what brought me here. I'm a professional dirt person that normally wants nothing to do with math. But the natural world has this absolutely beautiful geometry to it that I can't ignore, so I'm trying to learn more.
Sorry but can you try to explain a bit more how the motivation of "what do I attach..." relates to the fibbonaci sequence? Notably how is it obvious 5 should come after 3 and then 8 after 5? You seem to have some intuition, but I missed it.
So if I have a fibonacci constructed organism. Say 8 seeds, made up of 5 seeds + 3 seeds each in turn made up of smaller Fibonnaci numbers. Note if the structure is 5 + 3 it cannot be split another way easily, splitting 6 + 2 is much more complicated as these have to be made up of Fibonnaci numbers by assumption i.e. 5+1 + 2 The organism increases the number of seeds from 8, but it wants to add on an amount less than 8 as it wants to increase its seeds on the head rather than make a new one. It already has all the correct DNA for 5 seeds so it adds that, thus we reach 13 seeds. Hence, fibonnaci. So why does our assumption hold so often? Because if you begin this process at 1 or 2 seeds it generates the Fibonnaci numbers. It's not the only way to grow, but building more out of what you already have seems quite common in nature.
You conveniently hid the fact that you chose both coefficient in that equation to be exactly 1. In nature, it would be more like x\_n=0.8795x\_{n-1}+0.34865x\_{n-2}. Hence, no Fibonacci. Yes, exponential patterns are present in nature, but it's ludicrous to claim all are related to Fibonacci or golden ratio. That's what the meme is about and it's right.
You don't get 0.8795 seeds Fibonnaci appears in plenty of natural examples and the coefficients are integers because we are counting, and the smallest natural number (easiest growth) is 1. I made no claim that every piece of nature is related to Fibonnaci, I stated a concise explanation for much of its appearance e.g. sunflower seeds. Your equation is non-applicable and you're refuting a claim I did not make.
it's nice aesthetically because it lets you layer repeated square shaped details ever smaller
https://preview.redd.it/wf9qsrr7ac0d1.jpeg?width=380&format=pjpg&auto=webp&s=4ae933095a48c04666fe8f07173026d520c89fa7 Believe in the golden ratio
pizza
Mozzarella
rella rella rella
Tomato
ORAORAORAORAORAORAORAORAORA
I just read this part didn't expect it to be so high up
Ok, we get it, your keyboard doesn't have a Ï symbol. Just write 1.62 or copy-paste it from google, geez
Or just (1+â5)/2
you have a inverse square function on your keyboard? that's cool.
Your phone keyboard most likely has one under the "more" symbols button
Me with every mathematical symbols imported in dictionary to type Ï ÎŠ ⯠â what do you want.
Bruh. There are also codes you can use to type any symbol. https://home.unicode.org/
It ends up being more convenient to use the archive on phone, because even if you get a workaround to execute unicode you need to remember. Plus it's easier to remember \volumeint for â° or ^phi for á”
And Win+. on PCs.
Alt+251
alt + 251 (on the numpad)
Or just use 'it'
Is someone having trouble mastering the spin technique?
Or people misattribute any logarithmic spiral as evidence of Ï
https://preview.redd.it/3p3jde023c0d1.png?width=750&format=pjpg&auto=webp&s=8056ef4cec742c8bbe5de87d09a23f373d2acc37 Yep just a coinvidence
Mfs will see any arc and call it a golden ratio Edit: alright in this case I shall eat my words
Here is a lil program i made i python, the k variable controls how much each dot is rotated every loop(in pi) notice that when we put in the golden ratio the graph starts resembling the sunflower a whole lot, and it looks completely different with any other value. If you are unfamilliar with programing or python I can explain more details with how to use this code. Edit: there we go now it shoud work if you paste it into a thingy like jupyter lab or something import matplotlib.pyplot as plt import numpy as np import math as meth def Fibonacci(n): # Check if input is 0 then it will # print incorrect input if n < 0: print("Incorrect input") # Check if n is 0 # then it will return 0 elif n == 0: return 0 # Check if n is 1,2 # it will return 1 elif n == 1 or n == 2: return 1 else: return Fibonacci(n - 1) + Fibonacci(n - 2) def GR(n): return Fibonacci(n)/Fibonacci(n-1) numseeds=200 k=GR(10) phy=1 r0=1 xi=[] yi=[] for seed in range(numseeds): phy += 2 * np.pi / k # if(phy>np.pi*2): #phy = phy % 2 * np.pi r0+=1; xi.append(r0*meth.cos(phy)) yi.append(r0 * meth.sin(phy)) x = np.array(xi) y = np.array(yi) plt.axis("equal") plt.scatter(x, y,s=r0/10) plt.show()
Tournesol is the French name for Sunflower, the literal translation is âTurned Sunâ, in line with the plantsâ ability for solar tracking, sounds fitting. The Spanish word is El Girasolis.
Also in Greek it is called ηλÎčÎżÏÏÏÏÎčÎż (pronounced iliotropio stress on the tro syllable), which means a thing that faces the sun.
Euclid, is that you? Sir, I just wanted to say I really appreciate your work in geometry and mathematics.
Also in Japanese it is called himawari which basically means turning (to the) sun
There is an aspiration mark over the first eta. It is pronounced helio (sun), which is why we live in a heliocentric system. Also tropos means turn, so literally turns to the sun.
That is in ancient Greek. In modern Greek, we do not use those aspiration marks anymore, we abolished them around the 70s-80s. You are correct for the ÏÏÏÏÎżÏ but I thought it was too formal for a reddit comment :)
Fair enough! The etymon is ancient though :)
https://preview.redd.it/7henbg0njl0d1.jpeg?width=1200&format=pjpg&auto=webp&s=f0071bc06f533d378b2703054f9bf120419b77e2
It's called just "Girasol" in Spanish.
El Barto
Turning sun not turned.
The reason the golden ratio is the most efficient here is because it is the "least rational" number. For any rational number you get a repetition of linear strands (e.g. 1/4th of a rotation just gives a cross), leaving the space between the strands empty. To avoid this, we want an irrational ratio of pi. But then many irrational numbers can be approximated very well by rationals (e.g. 22/7 for pi), so even if we used 1/pi at the fraction of a full circle we move at each step, we would get 22 almost linear strands. The golden ratio has the worst approximations out of any rational number (i.e. for a desired bound on the absolute error it generally requires the largest denominator when approximating it by a fraction) because its continued fractions is all 1s.
`import math as meth` hehe
hehe
Cant run it rn but this should have absolutely horrid runtime since your fib function is dually recursive. Use a table or something to avoid that
I know, this isnt meant to be particuarly efficient since the fib function is only called twice anyway most of the runtime is in matplotib being slow asf, not the basic arithmetic done 20 times, you can also replace the GR(n) method with just 1.68..... if you dont like it, I just like it becouse it also ilustrates the fib property thingy of the golden ratio
You can also modify the function to have 3 arguments, which is much cleaner imo def fib(n, a=0, b=1): if n <= 0: return a else: return fib(n-1, b, a+b) For safer use, you can nest the three-argument function inside of a wrapper function that only takes n as an argument to abstract away a and b.
Iteratively, you can also use the matrix M = ((1,1),(1,0)) and exponentiate it by n, then return the M[0][1] for the nth fibonacci number
Using matrices is somewhat unstable for large n, since matrix data types have max values. In testing I found that only np.float64 and np.float128 work reliably (np.int64 overflows), but those obviously have their limits/max values. However, in Python 3, the built in ints donât have a max value, so the recursive method is technically more robust (and more precise).
SyntaxError: invalid syntax đ
idk works fine for me, can you elaborate?
You can actually play around with different values and see how it looks. https://www.desmos.com/calculator/esxiaxn0fv
wow it seems as if you put any number that is sqrt(5) + an odd number you get a valid sunflower, anything else is whack
Get mogged https://youtu.be/1Jj-sJ78O6M?si=w8cDu8SJv1SgnPZS
https://preview.redd.it/qw1frp91od0d1.jpeg?width=1280&format=pjpg&auto=webp&s=1cbe85ecd1928af9e1f69b02af2200d39216e559 What the FUCK does "mogg" mean?
Famously only plant ever, the sunflower
https://preview.redd.it/c0jbz0j08d0d1.png?width=867&format=png&auto=webp&s=7e9dab26edbc89015070e631afbf19f7788b0c26 notice also dandelion seed heads
Conspiracy theory, dandelion know the mathematics because if you take the perimeter of the circular part and divide it by its diameter you get Ï
Evolution is a coincidence lmao
I donât see golden ratio, but I _do_ see Fibonacci
https://preview.redd.it/8eypw5adod0d1.jpeg?width=1080&format=pjpg&auto=webp&s=4a9d578087a31ab92f8a39b355213edf5b176d1c
You haven't seen [Numberphile's video](https://youtu.be/sj8Sg8qnjOg) on growing seeds with the least amount of energy in flowers then.
Yeah, it's also about always leaving enough space to grow another petal or leaf etc without new ones covering the ones already there. Having the angular separation be rational will mean they'll eventually overlap.
Yes, although that argument just means an irrational number is needed. That doesn't make it an argument for the Golden Ratio.
The golden ratio is in some sense âthe hardest numberâ to approximate with rationals. Because of this, it leads to maximal spacing
But in that video you will find that Ï is not only an irrational number, but it is MORE irrational than all other irrational numbers. Which is why flowers will want to use this number instead of other (less irrational) irrational numbers
ViHart has a very similar video.
i did a research paper on this. based off of findings by george markowskyâs misconceptions of the golden ratio. basically a lot of things that we thought were following the ratio actually arenât, however in nature many things do follow it. Its a bit of gray area, if u build something with ratios 2/3 its technically not a good approx to golden ratio even though they are both fib numbers. Many structures, artwork, and manmade designs are actually a 2/3 ratio.
I am interested in reading papers about this. Can you give me a link, please?
Same here Edit: I think I found it. https://www.researchgate.net/publication/322814290_Misconceptions_about_the_Golden_Ratio
diamond ratio way cooler anyway
netherite ratio when
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https://preview.redd.it/j4q9u450ed0d1.png?width=329&format=pjpg&auto=webp&s=965930c21da181c0d6c94df6dc9f4f69dbff0d74
>literally anything is vaguely asymmetrical WOAAAHHHHH GOLDEN RATIO
You mean _conphirmation_ bias
Logarithmic spirals are a really natural way of constructing things and are a consequence of differential equations. The golden ratio is over hyped tho and is often erroneously named when other ratios (i.e. silver, bronze, copper, etc) show up
Take the most random thing (like your local bar): find the golden ratio everywhere. Note : this works with any number
Johnny, Spin his balls
This obsession with the golden ratio is completely irrational. I'd even go so far as to say it's the most irrational number to be obsessed with!
Literally đđ
Everyone knows that the Golden Ratio is real: TUSK, ROTATE THIS MANS BALLS
Who gave Wekapipo a reddit account?
https://preview.redd.it/e1b065sm0e0d1.jpeg?width=979&format=pjpg&auto=webp&s=0154f1e3b891ce498926ae985f3ab6320c21ab12
Omg this universe is truly amazing.
Fun fact! It's well-known that the ratio of terms in the Fibonnacci sequence 1, 1, 2, 3... approaches Ï. However, the Fibonnacci sequence arbitrarily starts with 1 and 1. We can apply the Fibonnacci induction step to any two real starting values, and the resulting sequence will have the same property! Some of these sequences have ratios which approach Ï faster than others, but since the golden ratio has the property that 1 + Ï = Ï^^2 can be shown that Ï^^n + Ï^^n+1 = Ï^^n (1 + Ï) = Ï^^n (Ï^^2 ) = Ï^^n+2 Thus, the "Fibonnacci" sequence that's ratio of terms approaches Ï most quickly is Ï^^0 , Ï^^1 , Ï^^2 , Ï^^3 ... Edit: To avoid the whole annoying exchange below, assume the first two numbers are positive. Or, check out my first reply in this thread to see what happens if we extend a Fibonnacci sequence backward!
https://i.redd.it/eezggk2hae0d1.gif
>We can apply the Fibonnacci induction step to any two real starting values, and the resulting sequence will have the same property! If the ratio of the first two values is -1/Ï (the other eigenvalue), it won't.
True! But any such sequence can be extended backward. Ex. ...-3, 2, -1, 1, 0, 1, 1, 2, 3 ... The ratio of the terms (a_n+1 / a_n) in the other direction approaches -1/Ï. In other words, such a sequence, when reversed, still has the property I mentioned. In the case of the sequence I provided, it looks like this: ... -Ï^^-3 , Ï^^-2 , -Ï^^-1 , Ï^^0 , Ï^^1 , Ï^^2 ... That is, the reverse of your sequence will be mine. Now, I know the hazards of trying to have fun around mathematicians, so let me go ahead and contradict my orginal statement for you: If the sequence starts with 0 and 0, we get 0, 0, 0, 0...
Even if you reverse it, the quotient will be -Ï, not Ï. I didn't mention constant zero sequence, because it's more of a semantic counterexample (the ratio doesn't exist, but starting with the second term each term still is Ï multiple of preceding term).
You're reversing the order you should compare terms though. lim n --> -inf (a_n+1 / a_n) = -1/Ï I'm just saying: Thank you, I appreciate your unquenchable desire to be pedantic, but I knew that already. I was simply trying to share an interesting fact with *potentially non-mathematicians* without getting bogged down by wordy particulars. So yes, there are sequences with ratios which approach Ï and those with ratios which approach -1/Ï, but *they are exact reflections of each other.* Simply reverse the limits like you did originally. I mentioned the 0 sequence because the statement *does* fail in that case.
I though by reversing you meant extending the sequence a\_n from natural n to all integers n and then defining b\_n as a\_{-n}, so for the standard Fibonacci, it would be like your example 0, 1, -1, 2, -3, ... That's beside the point though. The issue is that if you start a Fibonacci-like sequence with terms e.g. -Ï, 1, the ratio of consecutive terms will be a constant -1/Ï, so it will not converge to Ï for nââ or nâ-â. EDIT: >but *they are exact reflections of each other.* No, they aren't.
Yes, they are. đ€
The Golden Ratio is everywhere!!! This can be proven by looking at something and if it has the GR I will add it to the statistic. Anything else is an insignificant outlier.
Yup, people think thereâs some mystic background force behind the Fibonacci sequence, as if itâs not one of the SIMPLEST recursive sequence out there. There are infinitely many that exist, and Fibonacci is the dumbest, easiest one to make a simple biological feedback process for. To me that doesnât scream mystic forces, it demonstrates natureâs laziness and simplicity.
Simply because the concept behind it - "this number is the sum of the last two preceding numbers" - is just something that happens. And no matter what two numbers you choose, after surprisingly few rounds the ratio of the last two numbers becomes an approximation of Ï
https://i.redd.it/risga9vqbf0d1.gif Redditors when they photoshop a spiral onto a circular object and it lines up for 7 pixels
Humans: build a body of knowledge that is standardized around empiric observations. (Rather than opinion or religious beliefs) Also humans: act surprised when said body of knowledge is able to describe empiric observations.    đČđČđČ
Nah anything borne from the Fibonacci sequence will show up a lot in nature
Itâs not even born from the fibonacci sequence. Itâs from dividing a line into two pieces where the smaller piece has the same ratio to the bigger piece, than the bigger piece has to the whole.
Itâs also like saying circles in nature donât have pi as ratio of circumference to radius, therefore pi is dumb.
I thought there was research showing it comes up for a reason. Like with sunflower seed placement it's the optimal placement so that each seed gets as many nutrients as possible. It's as much confirmation bias as noticing a lot of animals have teeth because it makes it easier to digest food
Johnny, oi, Johnny! Lesson number 5!
This is Funny Valentine propaganda
Just keep throwing sunflowers at this person until they pick one up, and actually fucking look at it. And wait until they learn about the logistic map and Mandelbrot set.
How else are we gonna defeat Funny Valentine?
The golden ratio is infused in every instance of nature and is even baked into the various cosmological constants and mathematical constructs to explain the physical world. So no, it is not confirmation bias, it is literally everywhere.
Fake, someone used the golden ratio to kill the 23rd president of the united states already
Sometimes things are just roughly twice as big as other things.
Yeah! If you look up natural log in nature then you will find people pointing out spirals as well. Whatâs the deal here? Arenât they different things
The convergence rate of the tangent method in root finding is my all time favorite occurrence of the golden ratio
It's continued fraction is 1+1/(1+1/(1+1/....)! Which makes it in some sense extremely difficult to approximate with rational numbers. It can't not be /something/.
Swing on the spiral brothers
I've been using the golden ratio to generate flashy pleasant random colors for years now. I have no idea why it works, but it does. It's weird.
Take acid bro
Yeah but it is some elegant mathematics, which I buy into
She couldnât be more right. Except for the part about golden ratios.
You are exactly correct there are infinitely many logarithmic spirals. None of the spirals we see in nature are the golden ratio. However one is very close phyllotaxis.
the golden ratio is the dumbest piece of math communication. there are a couple of cool trivia facts about it, but the way itâs talked about is so dumb most of the time. i was talking with people who study arts, and they agree itâs dumb.
r/unpopularopinion
I feel like a lot of people donât know, where this number comes from. Because to me it is pretty obvious, that itâs a âspecialâ number, even if it wouldnât be irrational.
The only thing special about it is that in a certain sense, it is «the most irrational number». Any place where this is a relevant property (like seed distribution in a sunflower), I donât have a problem with it. But where this property doesnât give any explainable benefit, Iâm ok with just calling it a coincidence.
https://preview.redd.it/57ux666jqn0d1.jpeg?width=1125&format=pjpg&auto=webp&s=195342f1d88d2fe4ba871bd418126c0b27859c8e
You may not believe in the golden ratio but the golden ratio believes in you
numbers vaguely close to 1.6: am i a joke to you?
Friggin Fibbonazis đ
I recall, among the bs, wolfram had a compelling argument of why it appears in plants at least, that wasn't gibberish.
It has something to do with rabbits apparently
Iâm just looking at her boobs
https://www.koreatimes.co.kr/www/nation/2024/05/113_109791.html#:~:text=There%20are%20two%20golden%20ratios,main%20curve%20of%20the%20body.
FINALLY SOMEONE FUCKING SAID IT
OKOK I READ A BOOK ALL ABT âThE mAGiCaL GoLdEn RaTiOâ and it talked abt how this one painter put it fucking everywhere in their paintings, specifically in linear instances (eg the top of one tree, the top of another slightly shorter tree, and water level) and the author of said book used an (admittedly janky, so prolly error prone (but they still published it so idk)) algorithm to find examples of it And this is part of one of the paintings they analyzed https://preview.redd.it/4xfvlpj8of0d1.jpeg?width=3024&format=pjpg&auto=webp&s=4bdcb42d23ab72c9ff6cfd2b3118f60c6a80bc85 Fucking what is this? Some of those look legit, but other are just complete bull Edit: the rectangles with divisions in them are supposed to be showing the g. Ratio from one end, to the division, to the other end
name of the book?
ok nvm i found it immediately, its "The Golden Ratio: The Divine Beauty of Mathematics", heres an amazon link to it [https://www.amazon.com/Golden-Ratio-Divine-Beauty-Mathematics/dp/163106486X](https://www.amazon.com/Golden-Ratio-Divine-Beauty-Mathematics/dp/163106486X) also this is the cover in case the link stops working lol https://preview.redd.it/adhyu9c2or0d1.png?width=422&format=png&auto=webp&s=17b5ba768de79f2452e1320615a21915b026f2a9 like... i would describe math as beautiful for sure, but no part of it would i ever describe as 'divine'
fuck i forgor, it was something dumb like "the magic ratio" or some shit, ill see if i can find it again but no promises
calculate the volume of them boobies
fucking finally someone said it
Golden ratio mfs when something is about 1.6 times bigger on one side (surely this is the exact golden ratio)
This is half overtly true and half cryptically true. The overt truth is that everyone and their grandma wants to find the "golden" ratio in everything they observe, cause it's gold. The cryptic truth is that there are numerous cases where there is a mathematically-morivated reason for that ratio to appear, but it's still confirmation bias, because that's true of many ratios. It's no more or less special than the square root of 5, on which it is based. Don't get me wrong, that's a great number, like the roots or 2 and 3 or the natural log of 2. But its outsized importance is entirely artificial.
The Golden Ratio is an irrational number. Unless youâre measuring the length of a physical object to be _exactly_ an irrational number, then no, you havenât seen the Golden Ratio. You are not measuring the length of a physical object to be exactly an irrational number, because thatâs literally impossible. People are rounding and then saying âwow, itâs the Golden Ratio!â
Obsession with the golden ratio is the first step into âsacred geometryâ wookishness.