9:24 "So we conclude that 3 is irrational." Whoa, that's quite the jump there.

@hOREP245

6 жыл бұрын

quick maffs

@bobrobert1123

6 жыл бұрын

Root 3 dummy

@diamondsmasher

6 жыл бұрын

Seth Person settle down, don't be irrational

@Cerzus

6 жыл бұрын

Caught that as well

@cukka99

6 жыл бұрын

They fixed it in the closed captioning

Is it a coincidence that Numberphile talked about this as well at the same day? :)

@Mathologer

6 жыл бұрын

What else can it be ? :)

@hauslerful

6 жыл бұрын

Mathematician's conspiring? o.O

@flyingmonkeybot

6 жыл бұрын

It's those HI pod guys. CGP did the same to Jake of Vsauce, but no one owns the facts, except maybe Euler.

@flyingmonkeybot

6 жыл бұрын

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?

@TrimutiusToo

6 жыл бұрын

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...

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.

@Mathologer

6 жыл бұрын

Spot on :)

@mallowthecloud

6 жыл бұрын

Yep, exactly. That is what the spiral of a southern hemisphere cyclone looks like.

@redsalmon9966

6 жыл бұрын

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

@klausolekristiansen2960

6 жыл бұрын

Changing the direction of the golden spiral would be easy.

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.

@Mathologer

6 жыл бұрын

Spot on :)

@juicyclaws

6 жыл бұрын

yep, the image is horizontally flipped

@hugo3222

3 жыл бұрын

@@juicyclaws Actually, if it was "vertically" flipped, you won't see the earth., but only some stars or the moon.

@esajpsasipes2822

2 жыл бұрын

@@hugo3222 i think flipping is meant like mapping all pixels of a photo to other place along a line

@PASHKULI

2 жыл бұрын

yes, but it is not due the rotation of the earth...

every video you make is a work of art! please upload more ♥

@Nick-ui9dr

4 ай бұрын

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! 👍 😂👌

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.

@Mathologer

6 жыл бұрын

Spot on :)

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

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!

Why are so many people talking about logarithmic spirals all of a sudden?

@AttilaAsztalos

6 жыл бұрын

...maybe because today is... (drumroll) phri-day...? (okay, okay, I'll show myself out...)

@mandolinic

6 жыл бұрын

It's because logarithmic spirals prove the flat earth ;-)

@dhdydg6276

6 жыл бұрын

theyre good spirals brent

@tehjamerz

6 жыл бұрын

Mandolinic k.gif

@GTLx16

6 жыл бұрын

Mandolinic the exact opposite actually.

What about non-quadratic irrationals like pi and e? What are the properties of their spirals?

@Mathologer

6 жыл бұрын

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 :)

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. :)

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

Great explanation on continued fractions! It's interesting how you use geometric modelling instead of the more common algebraic proofs :)

Thanks for your videos sir. You are a kind human and great teacher. I love your use of visual devices in these videos.

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.

@Mathologer

6 жыл бұрын

Glad this worked for you :)

Once again, you have blown my mind in a way I never thought possible.

@Mathologer

6 жыл бұрын

:)

So, the golden ratio lies between 1 and sqrt(5).

@Mathologer

6 жыл бұрын

Never thought of (1+root 5 )/2 as an average. Nice :)

@1oo1540

6 жыл бұрын

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.

@LudwigvanBeethoven2

6 жыл бұрын

Duh!

@RazvanMihaeanu

6 жыл бұрын

Mathloger, every time when I see something on KZread 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...

I love your explanations!

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?

I love your videos. And i havent even watched this yet but i know im gonna like it!

I love your videos and since subscribing a couple months ago and because of you I have become very interested in mathmatics.

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!

This is one of the best math related videos I've ever seen

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).

@santolok7662

2 жыл бұрын

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.

@wyattstevens8574

8 ай бұрын

​@@santolok7662And minor is 10:12:15 (1/4 : 1/5 : 1/6) in the same way!

A thanks right from the heart to mathologer!!

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!!!

thank you and it got better towards the end. I enjoyed it a lot

The x solution is the golden ratio, the thing the numbers have in common is that they are all part of the Fibonacci sequence

@Mathologer

6 жыл бұрын

Which is also not a coincidence :)

@yakov9ify

6 жыл бұрын

Mathologer well I pretty much guessed it was the golden ratio once I saw the Fibonacci sequence :)

@digitig

6 жыл бұрын

Because the golden ratio comes up in the closed form expression for the n-th Fibonacci number, of course.

@yakov9ify

6 жыл бұрын

Tim Rowe well that much is obvious lol

@digitig

6 жыл бұрын

Probably obvious to anyone who gets this far in a Mathlogger comments section, anyway. Not to everyone. :)

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?

1:17 had my crackin' up! I love this channel.

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!!

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! ;)

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?

Nice video. Thanks for posting.

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.

Great video as usual, thanks!

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.

the cyclone is going the wrong way. they spin counter-clockwise in the northern hemisphere, and thats clearly iceland

@ragnkja

6 жыл бұрын

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.

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

@Mathologer

6 жыл бұрын

Exactly :)

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).

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😄)

Youve incorporated this into the MLC logo as well as your youtube pictograph. So interesting.

All numbers are terms of the fibonnaci series, and solutions are Phi and 1/Phi.

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.

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

This channel is so good... Your nickname should be Mr. Insight

@Mathologer

6 жыл бұрын

:)

Sir really i love your explanation..love from india🇮🇳

This really made my day!!!

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.

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.

awesome video! I wonder what happens with other roots and transcendent numbers

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."

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".

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".

@Mathologer

6 жыл бұрын

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 :)

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.

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.

17:55 The solution is the golden Ratio x1= (1+sqrt(5))/2 x2= (1-sqrt(5))/2

Pythagoras (Πυθαγόρας) is the first mathematician we know, that is "responsible" for some of these maths. Awesome vid, as usual.

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.

Thank as always :) I really appreciate it :)

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

@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.

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 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.

Irrational numbers are number which cannot be represented in a/b form. How did you write root(3) as a ratio of two integers?

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.

@Mathologer

6 жыл бұрын

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.

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?

Really cool and interesting!

Hmm, now im curious what would result from a rectangle who's aspect ratio is a transcendental number

Is there any relationship between the Golden Ratio and Tesla's 3-6-9 / Vortex?

12:02 Oh wait that's Euclid's Algorithm isn' it? For some reason I feel so happy to realize that lol

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.

This video was gold

Absolutely true that some people take things too far but the universe is geometric and multidimensional and there does seem to be reason to look into fractal and geometric emergence of the universe, life and consciousness. But im not a physicist I just like to mix philosophy and physics and geometry seems like a good overlap

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?

@Mathologer

6 жыл бұрын

Keep asking questions, but choose wisely when you ask them :)

@dlevi67

6 жыл бұрын

Quote Dante: "non ragioniam di lor, ma guarda e passa" (let us not talk about them, but look and move on)

what a coincidence right as brady and tony release a metallic spiral video you release a golden spiral video

Rational finite spiral what about -2/7? is the spiral for this number just the same as for 7/2?

Where did you get that shirt? I need it.

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)

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.

So, to get cubic/quartic/quintic/... irrationals, we need to go up in dimension to get repeating patterns? I'm not sure how a "spiral" would work there, but it seems promising.

@Mathologer

6 жыл бұрын

There is no really straightforward generalisation of all this. Having said that people have definitely investigated this question. Here is a bit of a writeup of what is known. en.wikipedia.org/wiki/Hermite%27s_problem

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.

Question about the irrationality proof of sqrt(3): What tells us that the square filling goes on infinitely, i.e., that there is no point at which filling in a square fills the area completely? (as it would be the case if the ratio A/B was deliberately chosen to be rational)

@trogdorstrngbd

5 жыл бұрын

Look at 12:14.

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.

something i have always wondered about the golden spiral -- is the portion of the spiral inside each square really a perfect quarter-circle, or just approximately? it seems to me that there is something 'asymmetric' about the curve if the radius of curvature abruptly changes every quarter-turn, but no such asymmetry is evident from the polar form. is it just because the reference point by which i'm judging the curvature changes every quarter-turn? if so, that's a nice property that any quarter-turn section of the golden spiral can be fit to a quarter-circle, do all logarithmic spirals share this property?

@stephenbeck7222

6 жыл бұрын

I think he answers your question in the first minute. The true golden spiral is a log curve which happens to be closely approximated by quarter circles.

@turtlellamacow

6 жыл бұрын

you're right, missed that somehow! thanks

Have I missed it being 'ratio day' today?

12:15 that’s just a visual version of the Euclidean algorithm. Very nice link

3:25 These visual proofs really drive home the irrational nature of these numbers; although, for the nth roots of any non-nth-power integers, there is a very simple proof that even I (who only took Intermediate Maths, in high school, and stopped there) could deduce, on my own; and which, I think, should be taught to everyone, in primary school (or left as an exercise to deduce, by themselves, for a more fulfilling/satisfying learning experience): Let n√k = a/b, where a & b are positive, coprime integers, and b ≠ 1. Now, (a/b)^n = (a^n)/(b^n). Because of the Fundamental Theorem of Arithmetic, a and b, being positive, coprime integers, have unique prime factorizations, with no overlap; say: a = d*e*f, and b = g*h*i. Then, our integer k = (a/b)^n = (a^n)/(b^n) = (d*e*f)^n/(g*h*i)^n = (d^n)*(e^n)*(f^n)/(g^n)*(h^n)*(i^n); which, as we can see, is also a ratio between 2 positive, coprime integers; since raising the factors to the nth power didn’t yield any new factors; it just gave us n copies of the existing factors. Therefore, (a^n)/(b^n) is a fraction, in its smallest possible terms; and, therefore, it cannot be an integer: (a^n)/(b^n) ≠ k -> a/b ≠ n√k. We, therefore conclude that no root of an integer can ever be a proper fraction; and so, any roots of integers; which are not integers, themselves; are, therefore, irrational. *Q.E.D. 𑀩*

@PC_Simo

10 ай бұрын

You can try this with various values, for k, n, a, and b; say, let: k = 31 n = 3 a = 22 b = 7 See, what you get 😉.

Love that shirt!

We missed u man.

@Mathologer

6 жыл бұрын

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. :)

@Mathologer

6 жыл бұрын

???

@15schaa

6 жыл бұрын

I can't believe an educated man such as yourself is fooled by this Australia myth.

The last image is flipped l-r. I don't know if that's what you meant with the word wrong though.

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?

Nothing to be squared of! His best t-shirt yet.

Excellent!!!

All coefficients are Fibonacci numbers , more precisely continuos Fibonacci numbers i.e 6,7,8,9,10and11th Fibonacci numbers...∞

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

@MathOratory

6 жыл бұрын

Last puzzle ... fibonacci series terms as coefficients so x = golden ratio, right ? Beautiful video indeed sir ...

@Mathologer

6 жыл бұрын

+MathOratory That's it :)

@Mathologer

6 жыл бұрын

+theo konstantellos Did you watch to the end? :)

@MathOratory

6 жыл бұрын

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 ... :)