The golden ratio spiral: visual infinite descent
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- เผยแพร่เมื่อ 14 มิ.ย. 2024
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So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc.
Here are some articles that debunk a lot of the golden spiral in nature nonsense:
Clement Falbo
web.sonoma.edu/Math/faculty/fa...
George Hart
www.georgehart.com/rp/replicat...
/ @georgehart-math
Keith Devlin
devlinsangle.blogspot.com.au/2...
Here is a very good website devoted to everything to do with the golden ratio and Fibonacci
www.maths.surrey.ac.uk/hosted-...
Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating).
Enjoy!
9:24 "So we conclude that 3 is irrational."
Whoa, that's quite the jump there.
quick maffs
Root 3 dummy
Seth Person settle down, don't be irrational
Caught that as well
They fixed it in the closed captioning
Is it a coincidence that Numberphile talked about this as well at the same day? :)
What else can it be ? :)
Mathematician's conspiring? o.O
It's those HI pod guys. CGP did the same to Jake of Vsauce, but no one owns the facts, except maybe Euler.
As for the picture at 21:36, I've never seen a hurricane hit the US that wasn't spinning in the opposite direction. Where was this photo taken, or is there some trickery here?
They were talking about logarithmic spirals, where each time you add just one square even if it doesn't fill the rectangle, while in this video it can be multiple squares depending on aspect ratio...
For the final puzzle, the land mass on the top left looks like iceland so this is the north hemisphere, hurricanes in the north hemipshere always go counter clockwise because of the rotation of the earth.
Spot on :)
yep, the image is horizontally flipped
@@juicyclaws Actually, if it was "vertically" flipped, you won't see the earth., but only some stars or the moon.
@@hugo3222 i think flipping is meant like mapping all pixels of a photo to other place along a line
yes, but it is not due the rotation of the earth...
21:35...
Well, it depends if the satellite image is from the northern or southern hemisphere. That island kind of looks like a mirrored Iceland, which would make sense, since that spiral is only cyclonic in the southern hemisphere. The image would have to be flipped for the spiral to by cyclonic in the northern hemisphere. And that is a low pressure system (hence the clouds), so it must be associated with a cyclone, not an anticyclone.
Spot on :)
Yep, exactly. That is what the spiral of a southern hemisphere cyclone looks like.
Oh it’s a mirrored image didn’t expect that
But now I think about it, flipped the image is easier than changing the direction of the cyclone
Changing the direction of the golden spiral would be easy.
every video you make is a work of art! please upload more ♥
Yeah! Not just art but science fiction... Rather more like math fiction. And I love the way he do transition between fiction and reality... Really a Satyajit Ray calibre movie! 👍 😂👌
So, the golden ratio lies between 1 and sqrt(5).
Never thought of (1+root 5 )/2 as an average. Nice :)
Mathologer I wonder if one could define a set of ratios as the whatever mean of 1 and root 5, and whether or not those ratios would have any interesting properties.
Duh!
Mathloger, every time when I see something on TH-cam about the Golden Ratio I always wonder why is not shown the relation between the right triangle (with sides 1 and 2 ...while the hypotenuse is √5) and the circle. That puts the sunflower seed spreading/growth into a new light...
Why are so many people talking about logarithmic spirals all of a sudden?
...maybe because today is... (drumroll) phri-day...? (okay, okay, I'll show myself out...)
It's because logarithmic spirals prove the flat earth ;-)
theyre good spirals brent
Mandolinic k.gif
Mandolinic the exact opposite actually.
Awesome, as always! :)
My guess would be that the fact about the greatest common divisor at 12:03 is due to the Euclidean algorithm (speaking of Greek mathematicians :) ). The construction of the spiral is essentially a visualisation of this algorithm, which is quite an efficient way of computing GCD.
Spot on :)
What about non-quadratic irrationals like pi and e? What are the properties of their spirals?
Actually e has a spiral with a nice pattern whereas pi's spiral a bit all over the place. Just have a look at the video I link to at the end which talks about this in terms of continued fractions :)
I love how you use geometry to explain things. My math skills are not what they used to be，but some of your videos really bring a smile to my face
The x solution is the golden ratio, the thing the numbers have in common is that they are all part of the Fibonacci sequence
Which is also not a coincidence :)
Mathologer well I pretty much guessed it was the golden ratio once I saw the Fibonacci sequence :)
Because the golden ratio comes up in the closed form expression for the n-th Fibonacci number, of course.
Tim Rowe well that much is obvious lol
Probably obvious to anyone who gets this far in a Mathlogger comments section, anyway. Not to everyone. :)
Even by the standards of your channel, this was an absolutely exceptional video. It's a masterful example of clear explanations. Awesome.
An elegant presentation of how elegant math(s) can be at times.
Wow! This was again a very amazing video by Mathologer. Math is so magic.
im a phinatic myself and was excited to see the debunk portion of the video. love the vids mathloger!
Great video - thanks! Keep up the great work! I've finally gotten around to learning about continued fractions, and came across the square-cutting algorithm about a month ago. It's such a beautiful way to visualize continued fractions. Your explanation here is clear and enjoyable. I am envious of kids today who have at their disposal such wonderful ways to learn and explore interesting topics in math early on.
Glad this worked for you :)
the cyclone is going the wrong way. they spin counter-clockwise in the northern hemisphere, and thats clearly iceland
In order to go clockwise in the northern hemisphere, it would have to be a high-pressure cyclone, but we don't generally get those on Earth.
I like this themed bunch of videos. This should happen more often. You all should talk to each other and do some sort of themed week as a collaboration
This is thematically so close to the last Numberphile videos that I wonder what your inspiration was.
I am not accusing you of ripping them off, otherwise your production speed would be amazing. Is something going on in the world of Mathematics that reinvigorated the fascination with the golden ratio?
Great explanation on continued fractions! It's interesting how you use geometric modelling instead of the more common algebraic proofs :)
Great video as always. There was an empty place for silver and other metalic ratios. Hope to see more on that. Also you promised a video on P and NP stuff.
Thank you for making this. It's my favorite video on this channel so far. :)
The square spiral for rational numbers is a great visualization of the Euclidean algorithm! Which explains why the rational square spirals must terminate and why the final square has side lengths of the gcd of the two sides. Great video!
Once again, you have blown my mind in a way I never thought possible.
:)
Thanks for your videos sir. You are a kind human and great teacher. I love your use of visual devices in these videos.
I love your videos and since subscribing a couple months ago and because of you I have become very interested in mathmatics.
This is one of the best math related videos I've ever seen
I love your videos. I would have loved to have you as my teacher when I was a younger student (which I'm not). I'll suggest my son, who is 17, to watch at your channel. Great job!!!
A thanks right from the heart to mathologer!!
At 6:53 you could've just extracted √3 out of the top to get √3(2-√3), then the bottom and the top would cancel out and leave you with √3.
Wow! Thanks to you, I have an idea for another visualization of musical consonances (besides Lissajous).
I will try to programmatically depict a smooth increase of the 1x1 rectangle to the size of 1x2 with "spiral squares".
One side (x1) is the frequency of the main sound. The other side (from x1 to x2) is the frequency of the second sound.
I hope it will show the difference between "good" and "bad" two-tones.
Just intonation dictates that:
1:1 - prima, unison.
1:1.33.. (3:4) - natural "fourth".
1:1.5 (2:3) - natural "fifth".
1:2 - octave (e.g. 440 Hz and 880 Hz simultaneously).
Other ratios are more dissonant. One of the most dissonant is the triton (1:√2).
Interesting to try to construct 3D spirals of three-tones combinations.
For example
Major chord is 4:5:6. That will be 1 x 1.25 x 1.5 3D-shape.
@@santolok7662And minor is 10:12:15 (1/4 : 1/5 : 1/6) in the same way!
I love your videos. And i havent even watched this yet but i know im gonna like it!
The fact that the side length of the smallest square is the greatest common divisor for the two numbers is related to the Euclidean algorithm. When we find the squares, we are dividing A by B until we're left over with a remainder, A - xB, where x is the quotient of A/B. Then we repeat the same process, dividing B by the previous remainder and generating a new remainder. After enough iterations of the process, we wind up with a division problem where the remainder is 0. Since every side length for every square prior to that was the quotient of the two previous side lengths, we know that the final side length fits evenly into every other side length. Since the process can't be extended any further, we know that that must be the smallest possible side length that fits evenly into all the other side lengths. Hence, the greatest common divisor.
Literally a combination of the topics in the 2 most recent Numberphile videos, but with a lot added and done in classic Mathologer style. I'm not complaining at all, just makes me a tad suspicious! ;)
loved the trump spiral
Had to be done although I am sure that it will earn this video some dislikes :)
Nobody makes a better spiral. It's yuge. It's the best spiral ever. That, I can tell you. Make spirals great again!
He's the only thing more irrational than Φ
Φ on him, I say.
I mean, he has a "very, very good brain", so it is only natural that it produces the best spiral ever.
Hi Mathologer, I love your videos.
What happens when we try making spirals with cubic, quartic root numbers? How does it change from the quadratic case? And then, how does it then change when we move on to quintic roots, as there is no general quintic root formula? Finally, and most importantly, how does the picture change when we used transcendental numbers?
The final square side’s length is due to the Euclidean algorithm which says that ( if a>b):
gcd(a,b)=gcd(a-b,b)
A great video!!
I love your explanations!
Have I missed it being 'ratio day' today?
1:17 had my crackin' up! I love this channel.
Great video as usual, thanks!
um dude idk if trumps hair can be described as nature
Funny story, my dad used to work on his property in Florida, and he swears that before he started running for office he had a lot less hair. So... not much natural about it.
All numbers are terms of the fibonnaci series, and solutions are Phi and 1/Phi.
Great video as always ! Really interesting visualization !
Random comment : As showing that any periodic spiral is related to a quadratic formula is simple but the reverse is hard, is it possible to create the cryptographic function from it?
thank you and it got better towards the end. I enjoyed it a lot
@Mathologer the picture is of Iceland (I believe), but is flipped. For some reason no one thought to just flip the spiral instead of the image.
One answer to final question: The spiral should not be not tangential to the lines where corners of squares meet. They are in your picture because you have drawn circular arcs rather than a logarithmic spiral.
Would this work for 3 dimensions as well? i.e. for cube roots? My first thought when I saw the infinite spiral was if pi could be drawn like that. Then I remembered it can't because pi is transcendental.
I got curious as to what sort of curve would be obtained if we start drawing from a non-quadratic irrational. And even more, would a transcendental irrational have some particular type of spiral? (I'm looking at you, e and pi).
Unimportant detail, but before yesterdays numberphile video about the silver ratio, in which @DrTonyPadilla mentioned A4 paper and you now mentioning A4, I thought this was a German only thing, especially as A5 is short for DIN A4 here (DIN being for German what ANSI is for the US, the German Institute for Standardization).
Maybe worth a global look: en.wikipedia.org/wiki/Paper_size#/media/File:Prevalent_default_paper_size.svg
Funnily Australia is uncharted land here. So is it blue, or did you just import A series Paper for your own usage? Or is it mixed in Australia?
Last, not least, I'll not judge which video covers the topic better. @DrTonyPadilla has a nice opener from the fingernail experiment, anyway, it's not an exclusive content war. It's nice to see a topic from multiple perspectives.
Wait, so you're telling me Gyro Zeppeli was lying to me?
maybe in their alternate universe it was true.
Youve incorporated this into the MLC logo as well as your youtube pictograph. So interesting.
Numberphile rivalry intensifies.
Mathologer ftw
Numberphile is for idiots, like Linus tech tips
You know, everyone hypes about the silver ratio taking two squares off to make a new silver rectangle...what about taking one square from the middle to get two similar rectangles? The sequence goes: 0,1,1,3,5,11,21,43... Where the silver ratio takes a"=2a'+a, this takes a"=a'+2a, and the ratio of successive terms approaches x=1+2/x; x is 2. The general formula for this newly expanded spiral follows a"=na'+ma, r=n+m/r, r²-nr-m=0, and r=(n±sqrt(n²+4m))/2, which for all except m looks just like the golden formula.
This channel is so good... Your nickname should be Mr. Insight
:)
Typo around 20:13
1/0.7320... = 1/(√3 - 1) = (√3 + 1)/2 = 1.3660...
16:29
"... one of the usual suspects, Leonhard Euler."
To quote 3Blue1Brown on this matter:
"It's often joked that in math formulas [and theorems] have to be named after the second one to prove them because the first is always going to be Euler."
Nice video. Thanks for posting.
Hmmm... Thinking about it, any two numbers who's numbers on the real number line are close together should have very similar patterns, so it might be possible to build a program that has a smoothly varying spiral pattern as the number slowly increases/decreases. This might be beyond my programming capabilities, but I certainly intend to try. I'll post here if I decide to finish and upload the output, if people are interested.
Final frame of video,
The cyclone is over north america but is rotating anticlockwise, ¿que no?
❤️
You make me miss the days in Mr. Olson's maths classes. Small rural school, he farmed & taught euclidean geometry, trig & calc. ❤️
Dankashane!
(it was Pennsylvania-Dutch country; so i'm sorry if i misspelled 'thank you'...i only ever heard spoken German😄)
11:50 So the connection between the GCD of A and B in the original rectangle, and the smallest side length in the square sequence, is based on the Euclidean algorithm for calcuting a GCD, and the division theorem.
The divison theorem is simple. For all integers a,b: a = qb + r, where 0B, then we compute the side length of the next square by using the Euclidean algorithm in disguise. First square has length B. The next square's length can be thought of as the remainder of A/B, because you can make some number of squares with B, say q of them, but once you can't make another square with length B, and the next square must have length r1=A-qB, where q is how many whole B lengths we can fit inside A... so it looks like the division theorem above. If we continue this process again, the next square will have length r2=B-wr1, with quotient w for the division of B by r1..
The Euclidean algorithm part arises because it operates on a theorem that states that GCD(A,B)=GCD(B,r1)=GCD(r1,r2)=...=.GCD(r_n-1, rn)=rn, where rn is the last nonzero remainder.
The sequence will terminate with some smallest side length rn, for the smallest square side length, but the Euclidean algorithm let us trace the equalities back up and realize that GCD(A,B)=rn
For 1920/1080:
1920 = 1*1080 - 840
1080 = 1*840 + 240
840 = 3*240 + 120
240 = 2*120 + 0
Then 120 = rn = GCD(1920,1080)
Thx Euclid
Puzzle 2 ... Euclid's method of long division of calculating HCF ?? Did he actually visualize it in this way .. I understood the method using the idea of factors .. But this geometrical similarity is beautiful
Last puzzle ... fibonacci series terms as coefficients so x = golden ratio, right ?
Beautiful video indeed sir ...
+MathOratory That's it :)
+theo konstantellos Did you watch to the end? :)
Loved it ... I always used to show it like ... f*a - f*b = f*(a-b) ... So common factor and all .. But I'll definitely try this for fun in my next class ... amazing sir ...
I was just thinking of something in the sqrrt(3) ... square direction thing ... Don't know if I'm observing too much into it ..
The sequence is 1,1,2,1,2,1,2,1,2,... right?
Now I was looking into the direction of the arrows and observed something ... (maybe not relevant) ...
but if from the first 1,1,2 we take out '1' from each number ... that is taking '1' square in each direction (the first 3 that is)...
Then the terms left in the series is again
1,1,2,1,2,1,2 ...
So it's like:
1,1,2,1,2,1,2,1,2,1,2,....
= (1,1,1) + (1,1,2,1,2,1,2,1,2,....)
Obviously, this will then go recursively ... I just sat with pen and paper to calculate the same for sqrrt (2) and sqrrt (5) ... Dunno if anything is there or just a coincidence .... but it's 3 ones taken out right ,, and it is rt(3) afterall
Sorry I didn't have the square root symbol in my keyboard ... :)
12:02 Oh wait that's Euclid's Algorithm isn' it? For some reason I feel so happy to realize that lol
12:05 Euclid!
When we divide the large rectangle (A x B) into a set of squares of equal size (B x B) and a smaller rectangle (B x A-B) at each stage, we are basically running a single iteration of the Euclidean algorithm, where the number of squares is the quotient and the new side length (A-B) of the small rectangle is the remainder.
The algorithm terminates when we have no remainder left, in other words, when we have found the largest square whose side length is a factor of both the length and breadth of the original rectangle! :D
Exactly :)
It's amazing how easy it is to find the faces you are looking for in clouds...
Sir really i love your explanation..love from india🇮🇳
I'm glad that I'm not the only one who rolls their eyes when someone forces a golden spiral onto an image to "prove" it's well-designed or "natural".
17:55 The solution is the golden Ratio x1= (1+sqrt(5))/2 x2= (1-sqrt(5))/2
About the final picture of the cyclone. It's what you already showed at the beginning with logarithmic spirals, that would fit even better. Looking at the dark spiral of the cyclone, the gap between cloudy regions, that crosses the edges of the rectangles and arcs bleed over. I assume a real logarithmic spiral will have a smooth change in curvature, not be pieced together from quarter-circle arcs, thus they don't fit in the square regions of rectangles with these specific ratios.
I know it's a bit late, but if I understood this right then I'm a happy guy now.
One of the things I've never liked about the traditional definition of irrational numbers is that it is defined by a negative quality. I.e. it cannot be written as a fraction of whole numbers. If irrational infinite spiral and rational finite spiral then suddenly there exists an equally valid _positive_ definition, namely that irrational numbers are those numbers who has an infinite descend spiral.
Pythagoras (Πυθαγόρας) is the first mathematician we know, that is "responsible" for some of these maths.
Awesome vid, as usual.
Really cool and interesting!
awesome video! I wonder what happens with other roots and transcendent numbers
Thank as always :) I really appreciate it :)
This really made my day!!!
You can go so much deeper than the golden ratio.
You can create quadratics for all quadratic irrationals. Let's say we make a quadratic from a continued fraction, like sqrt(3). We'll say x=1+1/(1+1/(1+x)). This is equivalent to x = (2x+3)/(x+2).
When a quadratic generated this way of the form x = (ax+b)/(cx+d) and the solution to the quadratic is sqrt(D), then a^2 - Dc^2 = +/-1 (Pell Equation!) and b^2 - Dd^2 = -/+D.
In our sqrt(3) example, 2^2 - 3(1)^2=1 and 3^2 - 3(2)^2 = -3.
Did you and Numberphile conspire to release φ-related videos at around the same time, or is it a coincidence? Either way, this is an interesting video that shows a neat visualization for the continued fractions that they discussed in their video.
Coincidence. This sort of thing happens more often than you would think. Actually never great if you come second as it happened to me with this video. After something happened to me and Infinite series, 3Bue1Brown, Infinite series and exchanged lists of upcoming videos for a while to avoid this from happening. Numberphile was never part of this though.
Irrational numbers are number which cannot be represented in a/b form. How did you write root(3) as a ratio of two integers?
12:15 that’s just a visual version of the Euclidean algorithm. Very nice link
Another amazing video that help to connect analytic or algebraic results with visual geometry... Thank you very much.
I think that if Erdös would be alive it could talk about "The Channel" as a complement to "The Book". In this sense many of your videos must be in "The Channel".
Nice thought. I'm definitely always trying find the proof in "The Book" and then make it even more accessible with a video like this :)
Hi I am big fan, mathologer . I love maths and discovering new stuff in math. I ask a lot of questions in the class but my whole class lauphs at me and my teacher scolds me for asking useless questions . what should I do?
Keep asking questions, but choose wisely when you ask them :)
Quote Dante: "non ragioniam di lor, ma guarda e passa" (let us not talk about them, but look and move on)
This video was gold
Your video makes me want to go back in time to the school of Pythagorea in Egypt, but I would probably be thrown from a boat showing them this stuff.
Hmm, now im curious what would result from a rectangle who's aspect ratio is a transcendental number
math - nothing two b squared of
This is really funny :)
2b^2(m8) - two be squared of m-eight, but that would be a bit too complex, since you have to assume 2b^2 for function
It’s Euclid’s algorithm to find gcd, substract the smallest from the biggest and repeat until you get 0, the last number is the gcd
How can you determine if a continued fraction will go on forever without becoming periodic? And/or how can you tell how long the period might be if it's a very long period?
I want to play around with these ideas. Is there a computer program that will generate these rectangles for me with varying ratios? I would like to study and create a similar proof that PI and e are Irrational. If I can find a tool, it would be better than graph paper, and allow faster study and more insights.
what a coincidence right as brady and tony release a metallic spiral video you release a golden spiral video
We missed u man.
Yes, just surviving the first semester here in Australia and just don't have much time for anything but all the mission critical stuff. Anyway, only two weeks of teaching to go, should have more time for Mathologer after that. :)
???
I can't believe an educated man such as yourself is fooled by this Australia myth.
The most amazing thing about irrational numbers, to me, is that aproximately ALL numbers are irriational, proportion-wise. Even though there are infinite numbers, most of them are irrational.
The two solutions of the quadratic equation shown are the golden ratio and the silver ratio and the integers are fibonacci numbers... Not exactly surprising, but still, that's pretty neat.
I absolutely love this video ! but am sort of totally unconvinced by the proof of irrationally using the impossibility of the infinite spiraling staircase. Basically why do the infinitely many squares shrink down to a point? This seems totally counterintuitive to me, but is it not? Am I wrong here? Or are our intuitions then somehow different in general for some reason I wonder, and why would that be, or how possible etc? Thx
Is there a way to extend the notion of logarithmic spirals from the real numbers to the complex numbers? I know complex numbers can be expressed through infinite fractions, so it could theoretically be possible. Unfortunately, the only way I can think of doing it would be to have a 4 dimensional output - 2 dimensions for the 'a' in the rectangle, and 2 dimensions for the 'b' in the complex number, which is frustrating.
My guess to the last puzzle is that it is mirrored, the storm should spin the other way around because it is on the southern hemisphere
:)
iI like how you called Euler "one of the USUAL SUSPECTS "
Is there an easy way to find where an infinite spiral "terminates" in the limit of an infinite number of squares? Presumably the X,Y location inside the original rectangle approaches some limit but I don't know if there's anything interesting there.
The Golden Ratio is no mathematical magic but a physical phenomenom. If you add an object to a system in balance and strive to retain balance in the expanded system, that can be expressed as (1 stands for system in balance, phi stands for new physical object): 1/phi = 1+phi. The ratio of forces between the system in balance and the new object should be equal to the forces of the system in balance + new object. Phi is the expression of how the universe retains balance.
@Mathologer - I'm sorry to be that guy, but.... You got the root 2 spiral wrong at 2:57 . The final curve of the spiral starts on the first square, whereas, in all the other two square sets, you start the curve on the second square. Is there something I am missing or is it drawn incorrectly?
I found zero links on this spiral of squares depiction about numbers other that φ. Once you google if you get "spiral of square roots". Are there any?
I just discovered the continued nested radical sqrt(3+sqrt(3+sqrt(3+... converges on the ratio constant of the sequence a(n) = a(n-1) + 3*a(n-2)