This is a corrected version of a video that I uploaded two days ago. After a critical error was discovered in the "Nature's Numbers" part of the video, I decided to scrap the original video, fix the errors and republish the video. If you already watched the original video, you can skip straight to 34:07 Nature’s numbers and just re-watch this chapter which contains all the main fixes. I've collected the comments on the original video in this pdf file www.qedcat.com/phyl/comments_collection.pdf If you've got access to Mathematica, you can download my Helicone lab here: www.qedcat.com/phyl/helicone.nb Just in case you are wondering, in the original video, my list of best approximations of pi did not fit in with my definition. Apart from being a major error, this also entailed some missed opportunities for good explanations. :( As usual, also check out the description of this video for more background information about this video.
The new material is well worth the (re)watch! Now I wonder why you get 3 improvements by 1 before 22/7, 8 improvements by 7 before 355/113 and - I believe - 147 improvements by 113 before whatever comes next. Followed by only 1 improvement next, twice, it looks like based on OEIS #A063673. Where do 3,8,147,1,1 come from (if correct)?
It makes me happy to see someone as famous / well-known as you acknowledge a mistake, own it, and correct it. As a fellow mathematician / professor, I genuinely appreciate the humility you have shown!
I honestly don't know any mathematicians who aren't willing to acknowledge mistakes. I mean it's always hard but it's part of what I like about math. I mean if you aren't making mistakes you aren't working on hard enough material and the nice thing about math is you can't just pretend that your mistake was correct.
@ that seems fortuitous for you. I have had many professors, both math and non-math professors, that have refused to acknowledge a mistake. In one instance, the mistake was on an exam…professor refused to acknowledge the mistake, so I wrote down the trivial counterexample, wrote down what the professor intended the problem to be, then solved the intended problem. I’m not insinuating that a majority won’t acknowledge a mistake, just that they exist and I conjecture that as one’s popularity rises, the willingness to acknowledge a mistake decreases.
@ Ohh that's fair. I guess I'm thinking of interacting with them as a colleague when I think back to being a student there were certainly profs who resisted admitting they were wrong to students.
For many years, and from the first time I knew that I should support this wonderful math channel!, I really feel happy that you're still going forward!
This video is a great introduction to the helicone numberscope. I appreciate the clear explanation of how it works. The visuals are very helpful in understanding the concepts. I'm definitely going to try making one of these myself. Thanks for sharing this fascinating project!
Dear Mathologer, I would like to repeat my question I asked in the previous version of the video. Will you please please please ever publish that "insane" (your word! Your promise even) video about the Galois theory. I've really been waiting for it for a very long time now. You promised it after all. You did that in your video about the cubibal polynominals that they didn't teach for over 500 years. Thank you foe keeping your promise in advance. Happy 2025, A.
@@Mathologer yes, I remember to have watched that one a long time ago. I just rewatched it. (It's now 2:41 am, I should be sleeping already). I am pretty sure I understood everything you did there, but to be sure I'll rewatch it a second time in a few days. But I'm really waiting for the "completely insane" (yes that was your real promise) Level 7 of this video. Galois theory. I love completely insane video's. It will, I already am sure about that now, probably be the most interesting and exciting masterpiece you've ever made as mathologer. All the best, A.
Most excellent follow-up to the Fibonacci Flowers and the Most Irrational number videos. Loved the tree bit on the helicone! I appreciate the work and time you took to correct your small mistake. Your dedication to accuracy and truth is a worthy endeavor with manifest positive impact, and while it may be a given for you to be that way, I'll just point out that many many people don't operate like that, and in my eyes, this is a huge win for you because being open to correction and then correcting it looks better and is better than never making mistakes, because human. ❤
I'm a level designer, I make levels for video games, and I use the golden ratio for similar reasons as described in the Nature's Numbers section. If you have a limited number of art assets to place in a level, it's important to get as much use as possible from each one. In situations where I need to place one asset multiple times, I rotate each one by the golden ratio times 360 degrees (or 222 degrees, which is close enough). Imagine a row of bushes along a street. If each bush is the same exact model and placed with the same rotation, small details that repeat on each bush will stand out, and the human brain is really good at seeing that sort of pattern. But if each bush is rotated in the horizontal plane by 222 degrees more than the last one, then as much 'new' side of each bush will face the player as possible.
@Mathologer great observation skills! 🏆 I hosted an event about 15 years ago where someone was showing off an Enigma machine, and we were allowed to touch and handle it. 😀
@@Mathologer Not Simon Singh, I have never had the privilege of meeting him. Nor James Grime, who I have met. I am in a pub, and don't have the details to hand - wouldn't be too helpful as I only knew their handle/nick. If you have a 3d printer I commend my friend Craig Heath's "Replica Enigma Machine Rotor" model on Printables.
@@Mathologer You named him after C.F.G? Of course you did so. My favorite mathematician after B.P. 😀 He is cute. He must have taken after his mother. 😁
That table at 37:31 really shows how outstanding 355/113 is as an approximation of pi. It is astonishingly good for a three digit denominator. . Here's a question: Is there any other famous irrational number that has a rational approximation with a three digit denominator that is anywhere near as good as this one? To decide this, we need to think about making some measure of the "goodness" of a rational approximation (q) of an irrational number (r). To start with, we should calculate the proportional difference, (r-q)/r. We then realize that we must take absolute values to ignore the sign of the difference. We could then take the reciprocal so we can express this amount as being accurate to within 1 part in N, where N=|r/(r-q)|. (I use N here to indicate nearness of the approximation.) If we care only for how accurate the approximation is, this is a good measure and the higher the value of N, the better the approximation. But, that would have us just choosing bigger and bigger denominators as "better" when really some preference should be made for smaller denominators as these are generally found sooner and with less effort than bigger denominators (and perhaps smaller denominators are easier to remember). So, we could apply a penalty for larger denominators by dividing the score, N, by the denominator, d. But is this a sufficient penalty? Perhaps a better measure is to divide by the square of the denominator. I will settle on having a "rational approximation goodness score" calculated as: RAGS = | r / (r-q) | / d² Here are some scores (N and RAGS) for approximations of π (I have used the fractions given in the table at 37:31) Rational N-score RAGS 3/1 22 22.188 13/4 29 1.811 16/5 54 2.152 19/6 125 3.480 22/7 2,484 50.704 179/57 2,530 0.779 201/64 3,247 0.793 223/71 4,202 0.834 245/78 5,541 0.911 267/85 7,549 1.045 289/92 10,897 1.287 311/99 17,599 1.796 333/106 37,751 3.360 355/113 11,776,666 922.286 52,163/16,604 11,801,038 0.043 Note: The fractions given by the convergents from the continued fraction representation of π are 3/1, 22/7, 333/106, 355/113 and then the following: Rational N-score RAGS 103,993/33,102 5,436,310,128 4.961 104,348/33,215 9,473,241,406 8.587 208,341/66,317 25,675,763,649 5.838 312,689/99,532 107,797,908,602 10.881 Here are some scores for rational approximations of √2 Note: The sequence is given by a/b → (a+2b)/(a+b) Rational N-score RAGS 3/2 16 4.121 7/5 99 3.980 17/12 576 4.003 41/29 3,363 3.999 99/70 19,600 4.000 239/169 114,243 4.000 577/408 665,857 4.000 1,393/985 3,880,899 4.000 3,363/2,378 22,619,537 4.000 8,119/5,741 131,836,323 4.000 19,601/13,860 768,398,423 4.000 It is interesting that while the nearness (N-score) increases at a rate that converges towards 3+2√2, the denominator increases at a rate that converges towards 1+√2. Since (1+√2)² = 3+2√2, the adjusted score (RAGS) converges to a constant (=4) as the sequence progresses. Here are some scores (N and RAGS) for approximations of the golden ratio φ: Rational N-score RAGS-score 3/2 14 3.427 5/3 33 3.697 8/5 90 3.589 13/8 232 3.629 21/13 611 3.614 34/21 1,596 3.620 55/34 4,182 3.617 89/55 10,945 3.618 144/89 28,658 3.618 233/144 75,024 3.618 377/233 196,419 3.618 610/377 514,228 3.618 987/610 1,346,270 3.618 1,597/987 3,524,577 3.618 2,584/1,597 9,227,466 3.618 4,181/2,584 24,157,816 3.618 6,765/4,181 63,245,986 3.618 10,946/6,765 165,580,143 3.618 In the case of approximations of φ the improvement in nearness (N-score) occurs at a rate that converges towards 1+φ = φ². Then, as the denominator increases at a rate that converges to φ, the RAGS-score also converges to a constant which happens to be 2+φ = 1+ φ². The approximations of φ and √2 are generated by formulae that create convergence to fixed multiples for increases in the denominator. There is also a fixed rate of convergence towards the rational, as evidenced by the continued fraction representations of φ and √2 being 1+1/(1+1/(1+1/(1+1/… and 1+1/(2+1/(2+1/(2+1/… respectively. I think it is fair that this scoring system gives these formulaic fractions equal standing. It is also worth noting that the RAGS for φ is less than for √2, which is consistent with φ being the "more irrational" number that is harder to approximate. I have also calculated the N-scores and RAGS for √3 = 1+1/(1+1/(2+1/(1+1/(2+1/(1+1/(2+1/… The continued fraction pattern is 1,2,1,2,… It shouldn’t be a surprise that the N-scores increase at a rate that converges to 2+√3 while the RAGS converges to an alternating sequence of 3 and 6, with the higher RAGS coinciding with the approximations that are slightly greater than √3. This is because the approximations that are greater than √3 have a proportionately smaller increase in denominator than those that are less that √3 - i.e. if you go from above √3 to below √3, the denominator has increased by more than the numerator to obtain a smaller fraction. The scoring for the approximations of π is certainly more interesting since the continued fractions are not in a fixed pattern and so the quality of the approximations relative to denominator as indicated by the RAGS varies considerably. And just to come back to it: How good is the 355/113 approximation for π? It is the absolute stand-out amongst those shown here. [Note - these calculations were done "quick and dirty" in a spreadsheet and so the values are likely inaccurate past 10 digits.]
Beautiful. Reminds me of the golden period of screensavers. Math is trying to tell us something. Now apply this to the wave viscosity of the boson field to illuminate fundamental particles.
I'm a huge nerd that lives in the hyper rigorous textbooks usually, but there's still something fun about just seeing things that are "just" kind of cool! There's always a gap between intuition and rigor, and here too. Knowing that you can find relations from ratios and divergence and all is cool on paper, but seeing the visual patterns that it creates is fun too. Thank you so much for all of the incredible content :)
'Six Eyes is another innate attribute inherited within the Gojo clan. It grants the bearer extraordinary perception, allowing them to see the flow of cursed energy. The constant influx of information can be overwhelming, leaving the bearer overstimulated and tired. Six Eyes is an extremely rare trait.'
at ~38:16, u say that 113, 106, 99, 92 are all spcial as they are all part of the 7 spirals. Yet, i see eight "7"s in yellow. What does that indicative of? 8 spirals? thanks,
Have another look at this diagram here: ruclips.net/video/_YjNEfZ0VqU/видео.htmlsi=HA0qdwud_LfzhHx1&t=1762 . It highlights one of the 7 spirals. What I am saying is that 113, 106, 99, 92 are ALL part of one of these 7 spirals (because as numbers they are 7 apart). In fact, it's easy to visually trace the spiral in question in this diagram. Does that make it clearer?
@@Mathologer Thanks for replying. So then, There's no significance to the observation that there are 8 yellowed "difference" numbers. Further, can I conclude that 57 is on the same spiral as 113? Thanks! (Your vids are AWESOME. PS: I think 3blue1brown does a looksy into this too. And relates his findings to Mandelbrot). All very very cool!
Since it's a repost it will be mostly regulars like yourself who'll be watching it. Makes it into an exclusive Christmas present for all of you who've been supporting Mathologer over the years. :)
@@Mathologer No, I did not know that, thanks for the info. Yesterday I learned that you are from Germany and refering to the german Wikipedia from Würzburg. "Die Welt ist ein Dorf"
@@Mathologer I'm a civil engineer who likes to "look beyond one's nose". For example, about 10 years ago I worked as a surveyor for two years to better understand their work. I also worked as a scientist at the university for six years. In my opinion, the main problem is always the interfaces.
Well done again Mathologer. I think that 22/7 is practical - most people with basic arithmetical skills can do this by hand or even in their heard. For example - if I have a circle radius 10cm - we have 22/7 * 100 = 2200 / 7 = 314 cm squared - good enough if you are icing a Christmas cake.
There's something extra special going on with those approximations. With pi, for example, Wolfram lists the following approximations. 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215,... First: if you check the spirals corresponding to these fractions, you'll notice they alternate between spinning left and spinning right. (I'm too lazy to say CW or CCW/ACW/whatever) This is why 333/106 is listed even though 355/113 gets even closer for pretty much the same bang for your buck: the spirals must alternate in direction. The reason for this will be clear by the end of this post. Next: these fractions also correspond to evaluating the continued fractions. Related to that point is how they appear on the microscope: 3/1 is the first good approximation, and at the beginning the spiral just looks like one arm. Then as you zoom in, the approximation loops over itself until a better approximation appears: 22/7. The seven arms. Then the arms continue to loop over and twist among themselves until the next one over lines up nicely. This becomes 333/106 and 355/113. Notice what this means on the algebraic side: 22/7 = (7 * 3 + 1) / (7 * 1 + 0) 333/106 = (15 * 22 + 3) / (15 * 7 + 1) 355/113 = (1 * 333 + 22) / (1 * 106 + 7) 103993/33102 = (292 * 355 + 333) / (292 * 113 + 106) 104348/33215 = (1 * 103993 + 355) / (1 * 33102 + 113) For starters, this comes directly from the continued fraction. But now the connection between the microscope and the continued fraction is more obvious. Given the sequence A1/B1, A2/B2, the next term A3/B3 would be: (M * A2 + A1) / (M * B2 + B1) The logic for the denominator is this: B2 represents how many spirals there were on the last iteration, and B1 how many spirals before that. When the spirals loop over, the spiral that approaches is labelled B1, and it comes over in tiny steps of B2. This also explains the alternating nature of where the spirals curve. B1 must be opposite to B2's spin. Idk it's midnight I'm yapping and idk if this is correct
Actually, 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215,... is the sequence I had in the original video. In the end this sequence not giving the full picture prompted me to pull down the video and make this corrected video :)
45:23 Rewriting my original puzzle answer for the reupload: In short, 1 1 0 1 1 0 (repeating) is the Fibonacci sequence mod 2. Why does this work? Each pair (or n-tuple) of natural numbers, both labeled k, is offset by a certain angle from the pair of 0’s which could be said wlog to not move. This angle is (0.618k)/2, or (0.618k)/n in the general case. Which branch of the spiral each integer will appear to correspond to is determined by which nth part of the circle its angle is closest to. (0.618k)/n ≈ m/n in the reals mod 1. This can simplify to 0.618k ≈ m mod n for some nonnegative integer m < n We can look at the first two values: 0.618 ≈ 1 1.236 ≈ 1 This is the start of the Fibonacci sequence. It’s easy to see that if two natural numbers a and b add to c, then the corresponding angles A and B add to the corresponding C. So we can ignore the precise angles and just use the rounded results to get the next one! With two branches, 1 + 1 = 0 (opposite + opposite = same), 1 + 0 = 1 (opposite + same = opposite), 0 + 1 = 1 (same + opposite = opposite) And the cycle repeats. (Not rigorous part) Now since these are Fibonacci numbers, as the sequence continues, approximations using these numbers as denominators will become better and better, so the angles will approach whole multiples of 1/n. There is no chance of this pattern being disrupted by accumulating errors. With three branches the pattern will be 1 1 2 0 2 2 1 0 repeating, where zero is a spiral of all the same alignment, and twos and ones are spirals of cycling alignment in different orders. With 4, the pattern is 1 1 2 3 1 0 repeating, with zero being same-alignment spirals, 1 and 3 cycling in different directions between the four, and 2 cycling between one alignment and its exact opposite.
Need some spikey ear rings? store.wolfram.com/view/misc/#spikey_kit Also, this is an interesting article writings.stephenwolfram.com/2018/12/the-story-of-spikey/
This video (and some of the referenced ones) always remind me that I really should get around to truly learning about continued fractions... one of these days!
Totally awesome. It is no surprise that when the Golden Ratio is involved that the Fibonacci numbers and Continued Fractions are lurking in the background. The fractional part of divergence angle, Δ, can be thought of as Δ mod 1. So each leaf is placed on a circle of circumference 1 (radius 1/(2*pi)) after a rotation of Δ mod 1. From Ergodic theory, the rotations will be periodic iff Δ mod 1 is rational. In the aperiodic case, when Δ mod 1 is irrational, the leaves will become dense on the circle as number of leaves go to ininity. It certainly makes sense that nature (biologic) would want to use "irrational" Δs, but what about Interstellar Space? Can the arms of spiral nebulae, be modeled with the helicone and if so, why?
Great ! I was very suspicious of the list of best fractional approximations of pi given in the first video, but I did not expect them to behave so nicely ! Thanks for the reupload :) By the way, do you know how to determine all best fractional approximations of a given number using its continued fraction expansion ? It looks feasible, and perhaps then one could prove that the patterns we observe for pi are true for arbitrary irrational numbers ! (ie, the list of the denominators of these fractions is a concatenation of finite arithmetic sequences (with possibly only one term, as for the golden ratio) whose ending terms are given by the odd order truncations of the continued fraction, and whose reason (or common difference ? that's the way we say it in french) is the last term of the previous arithmetic sequence) ! Plus it's been quite some time since your last video about continued fractions !
I doubt that, increased complexity of the PCB would probably negate any cost savings. And brightness isn't really a limiting factor anymore. The LED max brightness cheaply achievable today is beyond blinding.
@@jurajvariny6034 I actually had a look and from what I've seen the LED clusters in traffic lights don't use a fancy golden ratio inspired distribution. Too bad :(
Interesting. Two questions: 1. To what extent (if at all) does the 'degree of irrationality' depend on the base of the number system used? 2. What happens to this 'degree of irrationality' if an irrational base number system is used (for example, let's say we count in base Pi - as we do for the radian angle system - or base Phi)? Clearly when using base Pi, Pi is no longer irrational, likewise, when using base Phi, Phi is no longer irrational. But when using base Pi, is Phi still the 'most irrational' number of all? Thanks (PS - I am not a mathematician so forgive me if these are dumb / obvious questions to you).
I liked it so much, I watched it twice! I did kinda want to see what happened if you used other irrational numbers... like the decimal part of sqrt(2).
If you've got access to Mathematica you can download and play with my helicone lab www.qedcat.com/phyl/helicone.nb Also here is a bare bones online version of the microscope that runs in a browser demonstrations.wolfram.com/PhyllotaxisExplained/
I didn't rewatch it entirely, but I enjoyed the new part and I did play it entirely nevertheless; now I also commented, and I gave it a like, which I rarely do. Lets see if we can let THE ALGORITHM bite its own tail!
Thanks for the nice table (starting 36:30). It's a pity that nothing is said about the "monster fraction" (iirc) any more. (The old - erroneous - one is the one preceding the new one in the list if it were continued, I guess.)
That's how it's done. The original helicone leaves are laser cut with little 3d printed inserts in the middle that feature the pins and grooves that the pins fit into.
Helices are also often referred to as 3d spirals and so with lots of things in this video looking like sprials I think spirals was the better word for this video.
The Moravian influence in the southeastern US results in the Herrnhuter Stern being sold in Christmas stores and even through Walmart - I learned it can be properly mathematically named as a stellated rhombicuboctahedron.
The Moravian star has its own Wiki page. Worth checking out. They call it a Kleetope of a rhombicuboctahedron. Stellation is something slightly different :)
Golden angle relationship:We have a circle with an angle of 360 degrees and we want to divide this circle into two angles so that the ratio of these two angles is equal to the golden number. We assume that these two angles are a and b and b>a. a+b=2π , b/a=φ & φ=1+√5/2 b=a.1+√5/2 , b+a=2π => a(1+√5/2 +1)=2π a(3+√5/2)=2π a=720/(3+√5) => a=720/5.236 =137.50° 360°_137.50°=222.5° 222.5°÷137.50°=1.618... =φ
For a helicone with 6 colours ( 9:07 in the video) I don't think its a mess - I see 3 spirals of 2 alternating colours red/green, blue/yellow and turquoise/orange. I wonder if it is related to the fact that 6 is 2x a Fibonacci number. It also slopes in the same direction as the slope with the helicone with 3 colours. EDIT: I just got to 14:45 for the 16 spirals where you already discuss that 16 is twice 8! Those 16 spirals to me are 8 spirals of 2 alternating colours. Also the 10 spirals picture at 14:15 looks like 5 spirals of 2 alternating colours (turquoise/orange, lime/dark blue, greeny/purple, teal/dark orange, golden yellow/blue). I wonder if the picture for say 9 spirals would be nice spirals if the line followed 3 sets of 3 colours (3x a Fibonacci number). EDIT again: After looking at 15, I think the way to get nice spirals is for a number to be a multiple of 2 Fibonacci numbers: 1 Fibonacci number dictates how many spirals, the other Fibonacci number dictates how many alternating colours are needed. So for example, 5=5x1 - 5 spirals of 1 colour each, 6=3x2 - 3 spirals of 2 alternating colours each and then for say 15=5x3 - 5 spirals of 3 alternating colours each.
Well I also say ... :) Anyway, yes, there is quite a bit of structure to be discovered in the distribution of colors, mainly based on the fact that 2x3=6 and that both 2 and 3 are Fibonacci numbers. And most of this extra structure jumps out at you when you focus on the banding with two colors each of three double spirals corresponding to three colours and the banding with three colours each of the double spirals corresponding to two colours :)
You know that I am a mathematician and you know my office number and you know my real name and you seem to be a local. Challenge for you: Find my office. Shouldn't be hard :)
Reminds me a bit of 3Blue1Brown video - though that one went in different direction, but there was also a bit about "straight spirals" that if you zoomed out were not so straight anymore. And those occurred when you did use one of those Pi approximations. I do have to say that compared to some of your usual.... tough videos that require a lot of pausing - that one was easier. And while Fibonacci numbers being the best approximations of golden ratio is kind of their definition, I was wondering if the same wouldn't be true (just it wouldn't be the "best") for any sequence of numbers, where you add 2 positive integers to themselves and then add result with the higher of those integers and repeat for couple of steps. Then the 2 consecutive numbers divided by each other (bigger/smaller) will be close to golden ratio. I think in fact I might have learned it from your channel.
Yes, that's true. (Pretty much) for any sequence that grows like the Fibonacci sequence the associated sequence of ratios of consecutive terms converges to the golden ratio. Have a look at this ruclips.net/video/cCXRUHUgvLI/видео.htmlsi=7wnt7yMXEvHFnlTx&t=1547
@@Mathologer Yeah. This was what I was talking about. Though I thought I might have learned about it in Lucas numbers video... Still power of phi were also very, very interesting.
I wonder if nicely filling 3d volume by some kind of spiral is achievable by using similar algorithm. All the spirals here are 2d, the helicone is basically 2d surface too. Or maybe it's impossible but perhaps you can combine 2 spirals in 4d as there are two independent rotation axes?
There are space filling curves, that is, 1d curves that fill a 3d blob like a cube, for examples. However, none of these can be nice and smooth like a spiral :)
Why does your Christmas tree with only green and red appear to have 2 spirals? One spiral to the right and one spiral to the left. Is it similar to what takes place in a sunflower seed pattern shown latet in the video?
The double spirals are a bit hard to see, but I did do the necessary highlighting earlier on in that video. Have a look ruclips.net/video/_YjNEfZ0VqU/видео.htmlsi=XP9zDVxnN37CWaLa&t=487
Is geometry the basis of math? Hm. What about other metalic ratios? It took me a bit, but I did figure out why you only used the fractional part. I wonder if I should work the gold angle into my app. That is a lot of fun.
Thank you for asking. Frank Harr's Conversion App. It's an Android unit and ratio conversion app. The cute thing is that it does fractions, up to two units in both input and output and has a ratio converter so you can do kg per meter to pounds per yard (if that's your thing). In my plane angle section, I have things like pi radians, and diamiter parts (1/60 of a radian) and I have a miscilaious section. I could add a metallic angle section.
I was going to repost what i said last time - that the Moravian star looks to be the final stellation of the icosahedron, but looking into it, it is actually a Kleetope of a rhombicuboctahedron! (and the Wikipedia article had a mistake so i fixed that)
Getting into continued fraction territory :) Maybe check out the earlier Mathologer video on this topic that I mentioned and that I link to in the description of this video.
@@Mathologer Nein nein, ich habe Ihren Akzent erkannt und eine kurze Suche ergab, dass Sie aus Würzburg stammen. Daher hab ich das einfach auf Deutsch geschrieben. Studiere momentan Informatik in Aachen und bin ziemlich an so kleinen Einblicken in die (reine) Mathematik interessiert.
Alles klar, dann. Ich lebe schon seit vielen Jahren in Australien, und Wörter wie “Kommentarsektion” klingen für meine alten deutschen Ohren irgendwie nicht ganz natürlich. Viel Spaß beim Studieren! Ich habe ursprünglich auch Mathematik und Informatik in Würzburg studiert :)
Actually, √5 isn't that bad. Like all real quadratic irrationals, it has a periodic continued fraction, and its largest continued fraction term is 4. It's not like π, which has big numbers like 292 and 161 in its continued fraction expansion.
Yep, you are absolutely right. I was really just interested in showing another distribution of leaves associated with a recognisable number that is clearly not as optimal as the golden ratio one, at least for "small" numbers of leaves :)
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
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This is a corrected version of a video that I uploaded two days ago. After a critical error was discovered in the "Nature's Numbers" part of the video, I decided to scrap the original video, fix the errors and republish the video. If you already watched the original video, you can skip straight to 34:07 Nature’s numbers and just re-watch this chapter which contains all the main fixes.
I've collected the comments on the original video in this pdf file
www.qedcat.com/phyl/comments_collection.pdf
If you've got access to Mathematica, you can download my Helicone lab here: www.qedcat.com/phyl/helicone.nb
Just in case you are wondering, in the original video, my list of best approximations of pi did not fit in with my definition. Apart from being a major error, this also entailed some missed opportunities for good explanations. :(
As usual, also check out the description of this video for more background information about this video.
Hi! the pdf file link returns a 404 page :(
@@DrFortyTwo Try it again now.
fixed 👍
The new material is well worth the (re)watch! Now I wonder why you get 3 improvements by 1 before 22/7, 8 improvements by 7 before 355/113 and - I believe - 147 improvements by 113 before whatever comes next. Followed by only 1 improvement next, twice, it looks like based on OEIS #A063673. Where do 3,8,147,1,1 come from (if correct)?
Your office is like a toy shop 😀
Once again I am rewarded for my procrastination: I never saw the original.
"Hard work often pays off after time, but laziness always pays off now".
Well hello there fellow engineer :p
It makes me happy to see someone as famous / well-known as you acknowledge a mistake, own it, and correct it. As a fellow mathematician / professor, I genuinely appreciate the humility you have shown!
I honestly don't know any mathematicians who aren't willing to acknowledge mistakes. I mean it's always hard but it's part of what I like about math.
I mean if you aren't making mistakes you aren't working on hard enough material and the nice thing about math is you can't just pretend that your mistake was correct.
@ that seems fortuitous for you. I have had many professors, both math and non-math professors, that have refused to acknowledge a mistake. In one instance, the mistake was on an exam…professor refused to acknowledge the mistake, so I wrote down the trivial counterexample, wrote down what the professor intended the problem to be, then solved the intended problem.
I’m not insinuating that a majority won’t acknowledge a mistake, just that they exist and I conjecture that as one’s popularity rises, the willingness to acknowledge a mistake decreases.
@ Ohh that's fair. I guess I'm thinking of interacting with them as a colleague when I think back to being a student there were certainly profs who resisted admitting they were wrong to students.
The number of ‘toys’ in your office is astounding. Always amazed at your presentations.
I'm doing my part! 😄
Thanks!
🪖
Would you like to know more?
@@jay_13875 Who knows the citation ist a man of culture
Gotta get that comment engagement tsfrfromm
For many years, and from the first time I knew that I should support this wonderful math channel!, I really feel happy that you're still going forward!
This video is a great introduction to the helicone numberscope. I appreciate the clear explanation of how it works. The visuals are very helpful in understanding the concepts. I'm definitely going to try making one of these myself. Thanks for sharing this fascinating project!
If you end up making a helicone please e-mail me a photo :)
Thank you for the corrected version!
Dear Mathologer,
I would like to repeat my question I asked in the previous version of the video. Will you please please please ever publish that "insane" (your word! Your promise even) video about the Galois theory. I've really been waiting for it for a very long time now. You promised it after all. You did that in your video about the cubibal polynominals that they didn't teach for over 500 years. Thank you foe keeping your promise in advance. Happy 2025, A.
I’d love to hear aboeut Galois theory
I have not forgotten :) Did you already watch this video ruclips.net/video/O1sPvUr0YC0/видео.htmlsi=tygWcAiia7LcCBtc ?
@@Mathologer yes, I remember to have watched that one a long time ago. I just rewatched it. (It's now 2:41 am, I should be sleeping already). I am pretty sure I understood everything you did there, but to be sure I'll rewatch it a second time in a few days. But I'm really waiting for the "completely insane" (yes that was your real promise) Level 7 of this video. Galois theory. I love completely insane video's. It will, I already am sure about that now, probably be the most interesting and exciting masterpiece you've ever made as mathologer.
All the best, A.
So many things to talk about, so little time. I actually looked at Galois theory again just before Christmas. Eventually it will happen :)
A great math video to start the new year!
Most excellent follow-up to the Fibonacci Flowers and the Most Irrational number videos. Loved the tree bit on the helicone!
I appreciate the work and time you took to correct your small mistake. Your dedication to accuracy and truth is a worthy endeavor with manifest positive impact, and while it may be a given for you to be that way, I'll just point out that many many people don't operate like that, and in my eyes, this is a huge win for you because being open to correction and then correcting it looks better and is better than never making mistakes, because human. ❤
I'm a level designer, I make levels for video games, and I use the golden ratio for similar reasons as described in the Nature's Numbers section. If you have a limited number of art assets to place in a level, it's important to get as much use as possible from each one. In situations where I need to place one asset multiple times, I rotate each one by the golden ratio times 360 degrees (or 222 degrees, which is close enough). Imagine a row of bushes along a street. If each bush is the same exact model and placed with the same rotation, small details that repeat on each bush will stand out, and the human brain is really good at seeing that sort of pattern. But if each bush is rotated in the horizontal plane by 222 degrees more than the last one, then as much 'new' side of each bush will face the player as possible.
Interesting !
Dear Mathologer, Karl and Lara, thanks for this new year's gift! 🎁 Have a great year. 😀
Thank you :)
Actually, I just noticed that in your logo you (?) appear to be holding an Enigma machine rotor. Is that right?
@Mathologer great observation skills! 🏆 I hosted an event about 15 years ago where someone was showing off an Enigma machine, and we were allowed to touch and handle it. 😀
I see. Was that Simon Singh?
@@Mathologer Not Simon Singh, I have never had the privilege of meeting him. Nor James Grime, who I have met. I am in a pub, and don't have the details to hand - wouldn't be too helpful as I only knew their handle/nick.
If you have a 3d printer I commend my friend Craig Heath's "Replica Enigma Machine Rotor" model on Printables.
Really enjoyed it the first time around and looking forward to a second. Is junior Mathologer a grad student or your son or maybe even both?
Junior Mathologer is my son Karl :)
@@Mathologer He must have had an amazing childhood in mathematical wonderland.
@@Mathologer You named him after C.F.G?
Of course you did so. My favorite mathematician after B.P. 😀
He is cute. He must have taken after his mother. 😁
@@harat-xwb Actually he's a Karl not a Carl :)
if Karl brings all this toys as dowry, I would like to marry him 😂😂😂😂😂
Perfection deserves attention...
Another amazing math adventure. Thank you and Merry Christmas, Mathologer!
Glad you enjoyed this particular maths adventure. Sadly not many people will get to see it :)
@@Mathologer Sorry to hear about the reupload. Still, it is there for completists (like myself) to stumble upon later on. Best, Floyd
In the end I am very happy that I fixed the video properly.
@@Mathologer The only way to live. Thank you, once again!
Thank you for releasing a corrected version despite the obvious drawbacks!
Thanks!
Thank you :)
This video contains so many interesting insights that just leads to more questions. 🎉 thank you for your fabulous work.
Glad you enjoyed it!
You can also draw the spirals by connecting a dot to next closest dot (by euclidean distance), then another spiral is to second closest dot, etc.
To some extent this is exactly what happens when we connect the dots into spirals in our minds :)
That table at 37:31 really shows how outstanding 355/113 is as an approximation of pi. It is astonishingly good for a three digit denominator.
.
Here's a question: Is there any other famous irrational number that has a rational approximation with a three digit denominator that is anywhere near as good as this one?
To decide this, we need to think about making some measure of the "goodness" of a rational approximation (q) of an irrational number (r). To start with, we should calculate the proportional difference, (r-q)/r.
We then realize that we must take absolute values to ignore the sign of the difference.
We could then take the reciprocal so we can express this amount as being accurate to within 1 part in N, where N=|r/(r-q)|. (I use N here to indicate nearness of the approximation.)
If we care only for how accurate the approximation is, this is a good measure and the higher the value of N, the better the approximation.
But, that would have us just choosing bigger and bigger denominators as "better" when really some preference should be made for smaller denominators as these are generally found sooner and with less effort than bigger denominators (and perhaps smaller denominators are easier to remember).
So, we could apply a penalty for larger denominators by dividing the score, N, by the denominator, d. But is this a sufficient penalty? Perhaps a better measure is to divide by the square of the denominator. I will settle on having a "rational approximation goodness score" calculated as:
RAGS = | r / (r-q) | / d²
Here are some scores (N and RAGS) for approximations of π
(I have used the fractions given in the table at 37:31)
Rational N-score RAGS
3/1 22 22.188
13/4 29 1.811
16/5 54 2.152
19/6 125 3.480
22/7 2,484 50.704
179/57 2,530 0.779
201/64 3,247 0.793
223/71 4,202 0.834
245/78 5,541 0.911
267/85 7,549 1.045
289/92 10,897 1.287
311/99 17,599 1.796
333/106 37,751 3.360
355/113 11,776,666 922.286
52,163/16,604 11,801,038 0.043
Note: The fractions given by the convergents from the continued fraction representation of π are
3/1, 22/7, 333/106, 355/113 and then the following:
Rational N-score RAGS
103,993/33,102 5,436,310,128 4.961
104,348/33,215 9,473,241,406 8.587
208,341/66,317 25,675,763,649 5.838
312,689/99,532 107,797,908,602 10.881
Here are some scores for rational approximations of √2
Note: The sequence is given by a/b → (a+2b)/(a+b)
Rational N-score RAGS
3/2 16 4.121
7/5 99 3.980
17/12 576 4.003
41/29 3,363 3.999
99/70 19,600 4.000
239/169 114,243 4.000
577/408 665,857 4.000
1,393/985 3,880,899 4.000
3,363/2,378 22,619,537 4.000
8,119/5,741 131,836,323 4.000
19,601/13,860 768,398,423 4.000
It is interesting that while the nearness (N-score) increases at a rate that converges towards 3+2√2, the denominator increases at a rate that converges towards 1+√2. Since (1+√2)² = 3+2√2, the adjusted score (RAGS) converges to a constant (=4) as the sequence progresses.
Here are some scores (N and RAGS) for approximations of the golden ratio φ:
Rational N-score RAGS-score
3/2 14 3.427
5/3 33 3.697
8/5 90 3.589
13/8 232 3.629
21/13 611 3.614
34/21 1,596 3.620
55/34 4,182 3.617
89/55 10,945 3.618
144/89 28,658 3.618
233/144 75,024 3.618
377/233 196,419 3.618
610/377 514,228 3.618
987/610 1,346,270 3.618
1,597/987 3,524,577 3.618
2,584/1,597 9,227,466 3.618
4,181/2,584 24,157,816 3.618
6,765/4,181 63,245,986 3.618
10,946/6,765 165,580,143 3.618
In the case of approximations of φ the improvement in nearness (N-score) occurs at a rate that converges towards 1+φ = φ². Then, as the denominator increases at a rate that converges to φ, the RAGS-score also converges to a constant which happens to be 2+φ = 1+ φ².
The approximations of φ and √2 are generated by formulae that create convergence to fixed multiples for increases in the denominator. There is also a fixed rate of convergence towards the rational, as evidenced by the continued fraction representations of φ and √2 being 1+1/(1+1/(1+1/(1+1/… and 1+1/(2+1/(2+1/(2+1/… respectively.
I think it is fair that this scoring system gives these formulaic fractions equal standing. It is also worth noting that the RAGS for φ is less than for √2, which is consistent with φ being the "more irrational" number that is harder to approximate.
I have also calculated the N-scores and RAGS for √3 = 1+1/(1+1/(2+1/(1+1/(2+1/(1+1/(2+1/…
The continued fraction pattern is 1,2,1,2,… It shouldn’t be a surprise that the N-scores increase at a rate that converges to 2+√3 while the RAGS converges to an alternating sequence of 3 and 6, with the higher RAGS coinciding with the approximations that are slightly greater than √3. This is because the approximations that are greater than √3 have a proportionately smaller increase in denominator than those that are less that √3 - i.e. if you go from above √3 to below √3, the denominator has increased by more than the numerator to obtain a smaller fraction.
The scoring for the approximations of π is certainly more interesting since the continued fractions are not in a fixed pattern and so the quality of the approximations relative to denominator as indicated by the RAGS varies considerably. And just to come back to it: How good is the 355/113 approximation for π? It is the absolute stand-out amongst those shown here.
[Note - these calculations were done "quick and dirty" in a spreadsheet and so the values are likely inaccurate past 10 digits.]
Liouville numbers have incredible rational approximations :)
@@jurajvariny6034 Thanks for the reply. True on Liouville numbers, but they are rather dull and artificial compared to Pi and 355/113.
Mandatory rewatch everyone! Do your thing, share with your friends who dont care about math!!
I don’t know how I’m subbed or why it was pushed, but I’ll watch your content. I saw your community post to like, so I liked. Happy 2025
Great. According to RUclips you've been subscribed for four years. Must have liked something then :)
This video is even better than the last one, because you fixed that error! Also, cool spiral thingies.
Thanks for watching! I'm glad you liked the fix.
Beautiful. Reminds me of the golden period of screensavers. Math is trying to tell us something. Now apply this to the wave viscosity of the boson field to illuminate fundamental particles.
I'm a huge nerd that lives in the hyper rigorous textbooks usually, but there's still something fun about just seeing things that are "just" kind of cool! There's always a gap between intuition and rigor, and here too. Knowing that you can find relations from ratios and divergence and all is cool on paper, but seeing the visual patterns that it creates is fun too.
Thank you so much for all of the incredible content :)
'Six Eyes is another innate attribute inherited within the Gojo clan. It grants the bearer extraordinary perception, allowing them to see the flow of cursed energy. The constant influx of information can be overwhelming, leaving the bearer overstimulated and tired. Six Eyes is an extremely rare trait.'
I am sure Mathologer junior knows, but I didn't :)
I didn't finish watching the first time! So now I get to finish the corrected version. I am not disappointed! :-)
That's great, mission accomplished :)
Thanks for the reupload!
at ~38:16, u say that 113, 106, 99, 92 are all spcial as they are all part of the 7 spirals. Yet, i see eight "7"s in yellow. What does that indicative of? 8 spirals? thanks,
Have another look at this diagram here: ruclips.net/video/_YjNEfZ0VqU/видео.htmlsi=HA0qdwud_LfzhHx1&t=1762 . It highlights one of the 7 spirals. What I am saying is that 113, 106, 99, 92 are ALL part of one of these 7 spirals (because as numbers they are 7 apart). In fact, it's easy to visually trace the spiral in question in this diagram. Does that make it clearer?
@@Mathologer Thanks for replying. So then, There's no significance to the observation that there are 8 yellowed "difference" numbers. Further, can I conclude that 57 is on the same spiral as 113? Thanks! (Your vids are AWESOME. PS: I think 3blue1brown does a looksy into this too. And relates his findings to Mandelbrot). All very very cool!
Yes, the 8 does not have a specific meaning here. And, also yes, since 113-57=56 is divisible by 7, both 113 and 57 are on the same "7-"spiral :)
@@Mathologer Thank you!
Rewatch 100%. Liked. Already subscibed.
Congrats on your integrity.
Since it's a repost it will be mostly regulars like yourself who'll be watching it. Makes it into an exclusive Christmas present for all of you who've been supporting Mathologer over the years. :)
I'm watching every SECOND to find the fixes!! Good decision.🎉
Thank you very much for your work and greetings from Würzburg!
You know that I am originally from Veitshöchheim, right?
@@Mathologer No, I did not know that, thanks for the info. Yesterday I learned that you are from Germany and refering to the german Wikipedia from Würzburg. "Die Welt ist ein Dorf"
So, do you do anything math(s) related for a job ?
@@Mathologer I'm a civil engineer who likes to "look beyond one's nose". For example, about 10 years ago I worked as a surveyor for two years to better understand their work. I also worked as a scientist at the university for six years. In my opinion, the main problem is always the interfaces.
Reminds me of my dad who was also a civil engineer who knew the jobs of pretty much anybody he was dealing with better than they did :)
Well done again Mathologer. I think that 22/7 is practical - most people with basic arithmetical skills can do this by hand or even in their heard. For example - if I have a circle radius 10cm - we have 22/7 * 100 = 2200 / 7 = 314 cm squared - good enough if you are icing a Christmas cake.
Absolutely :)
Great video, thank you, and thank you for the uploaded correction!
Here to see the corrected edition!
The second half of this video reminds me of a 3blue1brown video about spirals and straight arms that appear when plotting primes in polar coordinates.
Yo!! Good to see your doing well!
Woah, I thought the Mathologer only lived in the white void of pure mathematics
No, I actually also have an office in the real world :)
@@Mathologer An office full of great toys! After that teaser I'd be interested to see videos about them
Coming up this year :)
There's something extra special going on with those approximations. With pi, for example, Wolfram lists the following approximations.
3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215,...
First: if you check the spirals corresponding to these fractions, you'll notice they alternate between spinning left and spinning right. (I'm too lazy to say CW or CCW/ACW/whatever) This is why 333/106 is listed even though 355/113 gets even closer for pretty much the same bang for your buck: the spirals must alternate in direction.
The reason for this will be clear by the end of this post.
Next: these fractions also correspond to evaluating the continued fractions.
Related to that point is how they appear on the microscope:
3/1 is the first good approximation, and at the beginning the spiral just looks like one arm.
Then as you zoom in, the approximation loops over itself until a better approximation appears: 22/7. The seven arms.
Then the arms continue to loop over and twist among themselves until the next one over lines up nicely. This becomes 333/106 and 355/113.
Notice what this means on the algebraic side:
22/7 = (7 * 3 + 1) / (7 * 1 + 0)
333/106 = (15 * 22 + 3) / (15 * 7 + 1)
355/113 = (1 * 333 + 22) / (1 * 106 + 7)
103993/33102 = (292 * 355 + 333) / (292 * 113 + 106)
104348/33215 = (1 * 103993 + 355) / (1 * 33102 + 113)
For starters, this comes directly from the continued fraction. But now the connection between the microscope and the continued fraction is more obvious.
Given the sequence A1/B1, A2/B2, the next term A3/B3 would be: (M * A2 + A1) / (M * B2 + B1)
The logic for the denominator is this: B2 represents how many spirals there were on the last iteration, and B1 how many spirals before that. When the spirals loop over, the spiral that approaches is labelled B1, and it comes over in tiny steps of B2.
This also explains the alternating nature of where the spirals curve. B1 must be opposite to B2's spin.
Idk it's midnight I'm yapping and idk if this is correct
Actually, 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215,... is the sequence I had in the original video. In the end this sequence not giving the full picture prompted me to pull down the video and make this corrected video :)
It got interesting when rational approximations came in to play. I'll have to rewatch. As always, thank you for bringing out the beauty in math.
Glad you enjoyed the video :)
Loved the new addition/debugging
Yes, in the end, everything fell into place in a wonderfully smooth way-much better than I had originally expected.
45:23 Rewriting my original puzzle answer for the reupload:
In short, 1 1 0 1 1 0 (repeating) is the Fibonacci sequence mod 2.
Why does this work?
Each pair (or n-tuple) of natural numbers, both labeled k, is offset by a certain angle from the pair of 0’s which could be said wlog to not move. This angle is (0.618k)/2, or (0.618k)/n in the general case.
Which branch of the spiral each integer will appear to correspond to is determined by which nth part of the circle its angle is closest to.
(0.618k)/n ≈ m/n in the reals mod 1.
This can simplify to
0.618k ≈ m mod n for some nonnegative integer m < n
We can look at the first two values:
0.618 ≈ 1
1.236 ≈ 1
This is the start of the Fibonacci sequence. It’s easy to see that if two natural numbers a and b add to c, then the corresponding angles A and B add to the corresponding C.
So we can ignore the precise angles and just use the rounded results to get the next one! With two branches,
1 + 1 = 0 (opposite + opposite = same),
1 + 0 = 1 (opposite + same = opposite),
0 + 1 = 1 (same + opposite = opposite)
And the cycle repeats.
(Not rigorous part) Now since these are Fibonacci numbers, as the sequence continues, approximations using these numbers as denominators will become better and better, so the angles will approach whole multiples of 1/n. There is no chance of this pattern being disrupted by accumulating errors.
With three branches the pattern will be
1 1 2 0 2 2 1 0 repeating,
where zero is a spiral of all the same alignment, and twos and ones are spirals of cycling alignment in different orders.
With 4, the pattern is
1 1 2 3 1 0 repeating, with zero being same-alignment spirals, 1 and 3 cycling in different directions between the four, and 2 cycling between one alignment and its exact opposite.
That's great (double great, really) !
The video so nice I liked it twice.
A fantastic video as always.
Really had a great time making this one. Especially programming and playing with the helicone lab was so much fun.
Perfect topic, thank you for the amazing video!
Glad you liked it :)
As promised, I put the like also for this new version
Thank you!
I will always have a soft spot for the Wolfram spiky
Need some spikey ear rings? store.wolfram.com/view/misc/#spikey_kit Also, this is an interesting article writings.stephenwolfram.com/2018/12/the-story-of-spikey/
So will I 😌.
This is a great video, thank you!
Pure math can be experimental! Thanks for pointing out that aspect of the joy of discovery.
This video (and some of the referenced ones) always remind me that I really should get around to truly learning about continued fractions... one of these days!
egyptian fractions are even more closely related
Yes, you should !!
@@pedrosaune Egyptian fractions, another topic on my to-do list :)
Wonderful to get to see your office, really 🥰
The plan is to do a few more Mathologer videos that start out in my office this year :)
Goat of youtube math channels.
Glad you think so :)
Good thing is, I never finished the first video, so I need to watch this one anyways!
Totally awesome. It is no surprise that when the Golden Ratio is involved that the Fibonacci numbers and Continued Fractions are lurking in the background. The fractional part of divergence angle, Δ, can be thought of as Δ mod 1. So each leaf is placed on a circle of circumference 1 (radius 1/(2*pi)) after a rotation of Δ mod 1. From Ergodic theory, the rotations will be periodic iff Δ mod 1 is rational. In the aperiodic case, when Δ mod 1 is irrational, the leaves will become dense on the circle as number of leaves go to ininity. It certainly makes sense that nature (biologic) would want to use "irrational" Δs, but what about Interstellar Space? Can the arms of spiral nebulae, be modeled with the helicone and if so, why?
Great ! I was very suspicious of the list of best fractional approximations of pi given in the first video, but I did not expect them to behave so nicely ! Thanks for the reupload :)
By the way, do you know how to determine all best fractional approximations of a given number using its continued fraction expansion ? It looks feasible, and perhaps then one could prove that the patterns we observe for pi are true for arbitrary irrational numbers ! (ie, the list of the denominators of these fractions is a concatenation of finite arithmetic sequences (with possibly only one term, as for the golden ratio) whose ending terms are given by the odd order truncations of the continued fraction, and whose reason (or common difference ? that's the way we say it in french) is the last term of the previous arithmetic sequence) !
Plus it's been quite some time since your last video about continued fractions !
We can definitely spot where you corrected yourself :)
Presumably modern traffic lights have similar 137.5 degree packing for max brightness/area? They are made of lots of tiny LEDs.
Not sure what you mean. What is being packed in a traffic light?
@@Mathologer lots of tiny LEDs in the modern ones
@@tim40gabby25 Interesting, I'll have a look :)
I doubt that, increased complexity of the PCB would probably negate any cost savings. And brightness isn't really a limiting factor anymore. The LED max brightness cheaply achievable today is beyond blinding.
@@jurajvariny6034 I actually had a look and from what I've seen the LED clusters in traffic lights don't use a fancy golden ratio inspired distribution. Too bad :(
please tell me there is absolutely no relation between golden angle and sommerfield/fine structure constant ☃️🌲⛄
Pretty sure that 1/your constant \approx 137 and the golden angle being approximately 137.5 is just a coincidence :)
Amazing! Thanks for this gem
I like all the tree toppers. 🌟 😊
But if your life depends on it which one would you choose?
THANKS FOR THE CORRECTION
Merry Christmas...🎄
…still loving the start mostly ❤
Interesting. Two questions: 1. To what extent (if at all) does the 'degree of irrationality' depend on the base of the number system used? 2. What happens to this 'degree of irrationality' if an irrational base number system is used (for example, let's say we count in base Pi - as we do for the radian angle system - or base Phi)? Clearly when using base Pi, Pi is no longer irrational, likewise, when using base Phi, Phi is no longer irrational. But when using base Pi, is Phi still the 'most irrational' number of all? Thanks (PS - I am not a mathematician so forgive me if these are dumb / obvious questions to you).
None of the reasonable measures of irrationality depend on the choice of a particular base :)
I liked it so much, I watched it twice! I did kinda want to see what happened if you used other irrational numbers... like the decimal part of sqrt(2).
If you've got access to Mathematica you can download and play with my helicone lab www.qedcat.com/phyl/helicone.nb Also here is a bare bones online version of the microscope that runs in a browser demonstrations.wolfram.com/PhyllotaxisExplained/
Nice video! Even better this time 😊
It took forever to fix yesterday, but I’m very happy with how everything came together in the end. It was definitely worth the effort!
I didn't rewatch it entirely, but I enjoyed the new part and I did play it entirely nevertheless; now I also commented, and I gave it a like, which I rarely do. Lets see if we can let THE ALGORITHM bite its own tail!
Thank you very much for that :)
Gladly rewatching!!😃
Thanks :)
Your office looks like a toy store. So cool.
Thanks for the nice table (starting 36:30). It's a pity that nothing is said about the "monster fraction" (iirc) any more. (The old - erroneous - one is the one preceding the new one in the list if it were continued, I guess.)
Yes, you just keep going in the table it's the 162nd entry :)
Hmm. That is challenging to make in wood. I'm assuming that if I put a pin in each layer with a stops 137.5 degrees apart mod 360 I should get a tree?
That's how it's done. The original helicone leaves are laser cut with little 3d printed inserts in the middle that feature the pins and grooves that the pins fit into.
Updated and error-free. Hooray!
Yes, I'll be able to sleep tonight :) Of course, even now there is still room for improvement, there always is.
Aren't these Helices rather than Spirals?
Helices are also often referred to as 3d spirals and so with lots of things in this video looking like sprials I think spirals was the better word for this video.
why did I stop watching these videos so many years ago?
Me encantaría ver la torsión de esas paletas. Gran video.
The Moravian influence in the southeastern US results in the Herrnhuter Stern being sold in Christmas stores and even through Walmart - I learned it can be properly mathematically named as a stellated rhombicuboctahedron.
The Moravian star has its own Wiki page. Worth checking out. They call it a Kleetope of a rhombicuboctahedron. Stellation is something slightly different :)
Flawless Video!
Nice video!
Golden angle relationship:We have a circle with an angle of 360 degrees and we want to divide this circle into two angles so that the ratio of these two angles is equal to the golden number.
We assume that these two angles are a and b and b>a.
a+b=2π , b/a=φ & φ=1+√5/2
b=a.1+√5/2 , b+a=2π =>
a(1+√5/2 +1)=2π
a(3+√5/2)=2π
a=720/(3+√5) => a=720/5.236 =137.50°
360°_137.50°=222.5°
222.5°÷137.50°=1.618... =φ
That's it :)
For a helicone with 6 colours ( 9:07 in the video) I don't think its a mess - I see 3 spirals of 2 alternating colours red/green, blue/yellow and turquoise/orange. I wonder if it is related to the fact that 6 is 2x a Fibonacci number. It also slopes in the same direction as the slope with the helicone with 3 colours. EDIT: I just got to 14:45 for the 16 spirals where you already discuss that 16 is twice 8! Those 16 spirals to me are 8 spirals of 2 alternating colours. Also the 10 spirals picture at 14:15 looks like 5 spirals of 2 alternating colours (turquoise/orange, lime/dark blue, greeny/purple, teal/dark orange, golden yellow/blue). I wonder if the picture for say 9 spirals would be nice spirals if the line followed 3 sets of 3 colours (3x a Fibonacci number). EDIT again: After looking at 15, I think the way to get nice spirals is for a number to be a multiple of 2 Fibonacci numbers: 1 Fibonacci number dictates how many spirals, the other Fibonacci number dictates how many alternating colours are needed. So for example, 5=5x1 - 5 spirals of 1 colour each, 6=3x2 - 3 spirals of 2 alternating colours each and then for say 15=5x3 - 5 spirals of 3 alternating colours each.
Well I also say ... :) Anyway, yes, there is quite a bit of structure to be discovered in the distribution of colors, mainly based on the fact that 2x3=6 and that both 2 and 3 are Fibonacci numbers. And most of this extra structure jumps out at you when you focus on the banding with two colors each of three double spirals corresponding to three colours and the banding with three colours each of the double spirals corresponding to two colours :)
Where at Monash is your office? Would be cool to pop in and say hi sometime!
You know that I am a mathematician and you know my office number and you know my real name and you seem to be a local. Challenge for you: Find my office. Shouldn't be hard :)
why 6(2x3) wouldn’t act as good as 16(2x8), and form spiral lines?
It does and shows up nicely for small numbers of leaves, but for larger numbers the overlaps obscure what is going on.
Reminds me a bit of 3Blue1Brown video - though that one went in different direction, but there was also a bit about "straight spirals" that if you zoomed out were not so straight anymore. And those occurred when you did use one of those Pi approximations.
I do have to say that compared to some of your usual.... tough videos that require a lot of pausing - that one was easier. And while Fibonacci numbers being the best approximations of golden ratio is kind of their definition, I was wondering if the same wouldn't be true (just it wouldn't be the "best") for any sequence of numbers, where you add 2 positive integers to themselves and then add result with the higher of those integers and repeat for couple of steps. Then the 2 consecutive numbers divided by each other (bigger/smaller) will be close to golden ratio. I think in fact I might have learned it from your channel.
Yes, that's true. (Pretty much) for any sequence that grows like the Fibonacci sequence the associated sequence of ratios of consecutive terms converges to the golden ratio. Have a look at this ruclips.net/video/cCXRUHUgvLI/видео.htmlsi=7wnt7yMXEvHFnlTx&t=1547
@@Mathologer Yeah. This was what I was talking about. Though I thought I might have learned about it in Lucas numbers video... Still power of phi were also very, very interesting.
It may well be that I mentioned this before, maybe in the Tribonacci number video.
I wonder if nicely filling 3d volume by some kind of spiral is achievable by using similar algorithm. All the spirals here are 2d, the helicone is basically 2d surface too. Or maybe it's impossible but perhaps you can combine 2 spirals in 4d as there are two independent rotation axes?
There are space filling curves, that is, 1d curves that fill a 3d blob like a cube, for examples. However, none of these can be nice and smooth like a spiral :)
Why does your Christmas tree with only green and red appear to have 2 spirals? One spiral to the right and one spiral to the left. Is it similar to what takes place in a sunflower seed pattern shown latet in the video?
The double spirals are a bit hard to see, but I did do the necessary highlighting earlier on in that video. Have a look ruclips.net/video/_YjNEfZ0VqU/видео.htmlsi=XP9zDVxnN37CWaLa&t=487
such a lovely video
Excellent 👍
Is geometry the basis of math? Hm.
What about other metalic ratios?
It took me a bit, but I did figure out why you only used the fractional part.
I wonder if I should work the gold angle into my app.
That is a lot of fun.
What app?
Thank you for asking. Frank Harr's Conversion App. It's an Android unit and ratio conversion app. The cute thing is that it does fractions, up to two units in both input and output and has a ratio converter so you can do kg per meter to pounds per yard (if that's your thing).
In my plane angle section, I have things like pi radians, and diamiter parts (1/60 of a radian) and I have a miscilaious section. I could add a metallic angle section.
Thanks for that :)
Any time.
I don't think I'll add it. It feels a bit specialized. Not that I don't have a lot of speciealized units. But do I need more?
Like the video!
I wondered why this popped up again when I had already watched it. Now I understand.
I was going to repost what i said last time - that the Moravian star looks to be the final stellation of the icosahedron, but looking into it, it is actually a Kleetope of a rhombicuboctahedron! (and the Wikipedia article had a mistake so i fixed that)
Kleetope, now there is a word I had not heard before :) Interesting.
Great content
if you consider
x = 3 (floor of π)
u = ceil(1/(π-x)) = 7
x += 1/u (x = 22/7)
u = ceil(1/(π-x)) = 113
x+= 1/u (x=553/113)
and so on
Getting into continued fraction territory :) Maybe check out the earlier Mathologer video on this topic that I mentioned and that I link to in the description of this video.
Aber klar doch :D Hier auch noch eine Aktivität in der Kommentarsektion
Dieser Kommentar auf Deutsch basiert auf einem auf Englisch verfassten Text (automatisch übersetzt). Liege ich da richtig?
@@Mathologer Nein nein, ich habe Ihren Akzent erkannt und eine kurze Suche ergab, dass Sie aus Würzburg stammen. Daher hab ich das einfach auf Deutsch geschrieben. Studiere momentan Informatik in Aachen und bin ziemlich an so kleinen Einblicken in die (reine) Mathematik interessiert.
Alles klar, dann. Ich lebe schon seit vielen Jahren in Australien, und Wörter wie “Kommentarsektion” klingen für meine alten deutschen Ohren irgendwie nicht ganz natürlich. Viel Spaß beim Studieren! Ich habe ursprünglich auch Mathematik und Informatik in Würzburg studiert :)
Isn't that desk too low?
It's one of those fancy motorised desks that can be used at any height even standing up.
Hello is this another mathologer video?
Yes it is :)
Thanks a bunch.
Thank you Sir
Actually, √5 isn't that bad. Like all real quadratic irrationals, it has a periodic continued fraction, and its largest continued fraction term is 4. It's not like π, which has big numbers like 292 and 161 in its continued fraction expansion.
Yep, you are absolutely right. I was really just interested in showing another distribution of leaves associated with a recognisable number that is clearly not as optimal as the golden ratio one, at least for "small" numbers of leaves :)