You may have heard it said that this is the closest thing to a "least rational" number. In a Numberphile video a few years ago, Ben Sparks said that the continued fraction being all 1s is why that is- the ratios of Fibonacci numbers are actually the closest approximations to it!
@@wyattstevens8574 Yeah I mentioned in the description of this video that a future episode will cover the cool irrationality-related traits of the golden ratio :)
Fib(n) = Round(Phi ^ n / SqrRoot(5)) ; Phi = GoldenRatio ~ 1.61803... This was always my favorite as instead of tracking the last 2 values to add together, you can pluck it out by index with this spigot algorithm, which meant if you needed the n'th Fibonacci number, the old way you had to iterate n times to get to it, but this way you can calculate it immediately with no iteration.
My mind is honestly blown right now. I love the way certain concepts seem to just weave themselves into another concept in so many different ways. Like, the golden ratio being in the Fibonacci sequence once is crazy enough, but _multiple times?!_ This is what I love about math, and I’m so glad there are passionate people like you out here making videos for people like me to enjoy and expand our knowledge of this wacky number world
I'm really liking the more clean-cut, organized, Combo Class. All the older episodes really triggered my cleaning OCD, and made it hard to concentrate.
The connections between phi and the square root of 5 are interesting. Obviously it's in one of the definitions of phi = (1+sqrt(5))/2, but also phi^3 = 2+sqrt(5); also, dividing a Lucas number by its corresponding Fibonacci number tends towards sqrt(5).
I think 5 in general is interesting from a phi point of view. For example every 5th number in the standard Fibonacci is divisible by 5. In the Lucas sequence no number is divisible by 5, but the Lucas Cassini identity, ((An)-1 * (An)+1) - ((An)^2 ), that is the product of two numbers either side of a number, minus the square of that middle number, equals + or -5. In general 5 is hiding inside any what Domotro calls "Fibonacci-esque" sequence which contains no integer multiple of 5, such as his own "randomish" example at 11:02, 2 56 58 114 172 ... Note that 58*172 = 3020, so the Cassini identity for this sequence, + or -3020, is a multiple of 5. On the other hand any Fibonacciesque sequence which does go up in 5s, such as 2 57 59 116 175 ... has a Cassini identity of 3131, a non-multiple of 5. I also found that one solution for 𝔁⁵ - 𝔁³ - 𝔁² - 𝔁¹ - 𝔁¹ - 𝔁⁰ = 0 is phi. Note the sequence of exponents is part of the Fibonacci sequence in reverse, beginning with 5. I don't think this works for any other number.
The continued fraction of sqrt(5) = [2; 4], which is twice the sqrt(2) = [1; 2]. Both these continued fractions belong to the very special class of continued fractions of sqrt(n^2+1). Next is sqrt(10). Aren't we beautifull! Writing these as path informations where each digits turns to L if former is R and to L if former is R, in alternating manner looks like this (< for L and > for R for better visuals): sqrt(2): > etc. which looks even nicer when divided to rows >< >< etc. sqrt(5): >>>>
@@yasin_karaaslan It's quite fashionable these days to denigrate the decimal system in favor of the duodecimal for example, and I agree it's about time to reverse the trend. I'm not sure how I'd invoke the Fibonacci sequence for this, even though the 10th term using the convention of starting 1 0 1 1 ... is 55, which is also the sum of the first 10 natural numbers. A more useful fact is that the first 4 natural numbers, 1 2 3 4, sum to 10. This means if you place a counter in one of the 4 quadrants of a cross hair grid, a giant + sign, say the top right and call it 1, move it to the bottom right and call it 2, bottom left 3, top left 4, then put another counter in the top right so now you have a representation for 5, and keep on moving and adding counters till all quadrants are occupied, you will then have a representation for 10. Another possible stage in decimal counting history could have been the tic tac toe (noughts and crosses) grid. A counter in each successive space would represent 1 through 9, with an empty grid representing 0.
Domotro, I really can't praise this channel enough. Watching your channel now ignites the same spark of interest and passion for these topics that Numberphile inspired in me over a decade ago. I've fallen in and out of academics (I'm currently on an "extended hiatus" before finishing my engineering degree), but your videos remind me of why I loved math when I was younger and why I should love it again. Thanks.
You can even express powers of the golden ratio in terms of Fibonacci numbers Golden Ratio^(n) = F(n) × Golden Ratio + F(n-1) φ⁰ = 0φ + 1 φ¹ = 1φ + 0 φ² = 1φ + 1 φ³ = 2φ + 1 φ⁴ = 3φ + 2 φ⁵ = 5φ + 3 φ⁶ = 8φ + 5 φ⁷ = 13φ + 8 φ⁸ = 21φ + 13 φ⁹ = 34φ + 21 φ¹⁰ = 55φ + 34 And yes, the 0 case does work, the negative 1st Fibonacci number is also 1. You can work through the Fibonacci sequence backwords by thinking of what has to be added to the current first digit to equal the second digit. To get 1 from 0, you have to add 1 to it. Then to get 0 from 1, you have to add -1 to it, and so on.
I love how you only need 21 and 13 to make a good approximation of the golden ratio (21/13). Which is of course are part of the Fibonacci sequence (0 1 1 2 3 5 7 13 21 ...) Which impresses me even more because less than 10 elemnts and we're already almost there.
@@cherylchui4510 144=12^2. And sqrt(13) is the first continued fraction with greater than one odd number of repeating digits. Haven't yet figured out what might be the pattern of such numbers. Maybe, probably, somebody has, but dunno.
It may be a little off topic but I'm so obsessed with this number I named my daughter after it(in a way) buy assigning numbers to letters (a:1, b:2, etc), for 1618 I got A,F,A,H and AFAH is her name.
OH wow I've only ever seen this channel in a popped out window at work I didn't realize how small it was. this is easily 100k views a video content. thank u for the knowledge, super interesting video
I'm very much looking forward to your new series on the golden ratio and the intricate connections between the Fibonacci and Lucas numbers. It will be interesting to see whether you explore polynomial expressions for F[2n], F[3n], F[4n], etc., and L[2n], L[3n], L[4n], etc., whose coefficients can be derived from Pascal's triangle and a simple variation on it.
A neat observation with the randomly chosen fibonacci-esque sequence is how quickly the first couple digits begin to resemble the traditional fibonacci sequence
When Brady from Numberphile generated a Fibonacci-esque sequence with 2 random numbers, it ended up being known as the "Brady numbers" and it's even in the OEIS. So we should call the sequence from this video the "Domotro numbers".
I like the idea, but let’s save the name “Domotro numbers” in case I discover something extra unique. If this sequence needs to be referenced for some reason it can be called Domotro’s Randomish Fibonacci-esque sequence
What about a number which added to its reciprocal equals that number squared? x + 1/x = x² ? The one real solution (according to Wolfram Alpha anyway) is 1.4656. This number is the ratio constant in a sequence known as Narayana's Cows, just as the Golden Ratio is in the Fibonacci sequence. The difference is that every number is added not to the next but to the next-but-one. My own name for it is the Bovine Ratio, designated by the Greek character μ, pronounced "moo". The equation μ,+ 1/μ = μ² reads "Mummy moo plus baby moo equals moo squared".
I discovered the golden ratio at age 14 by looking for the number with the property 1/x = x-1. I also discovered that if you had a ravigneaux planetary gearset with a ring gear that is the golden ratio times larger than the sun gear (actually, both sun "gears" since there are two of them), you could make a four speed gearbox with ratios of 2.618, 1.618, 1.000, 0.618, and a reverse ratio of -1.618. (side note, I realize you can't make an irrational gear ratio with finite tooth count gears, but either approximate with large numbers or use discs with a high coefficient of friction)
This was filmed by my main camera guy Carlo. Some episodes have been filmed by other people too (who I name in the brief credits at the end of each episode)
I'm disappointed that you erased the 1s and replaced it with phis, instead of just drawing a 0 over the 1😁 8:11. I'm looking forward to the next episode on the golden ratio
Dude. I'm freaking out. I love your excitement regarding this and I feel as excited. I just figured out a thing I didn't know about Fibonacci numbers that relates to all this. Will share in discord
Lmao, I currently work at a bingo hall so the bingo ball cage was a bit of a trip, lmao. Since I've been working there tho I've realized theres a lot of complicated maths to do with bingo odds tho and so Ive been thinking a lot about combinatorics recently :) .
Two more facts about Phi: The Greatest Common Denominator between any consecutive fibonacci numbers is always 1. If you want to find the nth fibonacci number without calculating all previous numbers, the expression is ((phi)^n-(1/(phi))^n)/(sqrt(5)) Despite the equation being almost entirely made up of irrational numbers, an integer input will always produce an integer output.
That was great! .. reminds me of a numberphile with Ben Sparks who asserted that the golden ratio was "the most" irrational number since no numbers in the continued fraction are large.. in fact all ones , so it sorta threads that needle between rational numbers with the most room to spare on each side. There was a cool illustration with a graphic simulating a flower turning a certain amount each time it places another seed.. cool to see some other irrational numbers weren't so great because they're well approximated early on by a ratio (like pi had 3 spiral arms with wasted space ... eventually 22 spiral arms, etc.)
Yeah I love that topic. In the description of this vid I mentioned that a future episode will cover special traits the golden ratio has regarding irrationally
Amazing! … was just reading today about quadratic residue congruences for Fibonacci and Lucas numbers … would love if u could do an episode about quadratic reciprocity
Every time I think I've heard everything there is to know about the golden ratio, someone comes along and multiplies the amount I know by the golden ratio.
All Fibonacci-esque sequence is basically the same, meaning once you find a formula for one, you can easily have the formula of any other. 0, 1, 1, 2, 3, ..., f(n)=f(n-1)+f(n-2) 0, u, u, 2u, 3u, ..., uf(n) v, u+v, u+2v, 2u+3v, 3u+5v, ..., uf(n)+vf(n+1) a, b, a+b, a+2b, 2a+3b, 3a+5b, ..., (b-a)f(n)+af(n+1) = af(n-1)+bf(n)
This channel is only a little more than 1 year old so I’m happy with its growth so far. Hopefully will continue to grow! Also my other “bonus” Domotro channel actually has more subscribers just because of all the shorts I’ve posted there
You must be a first time viewer. He wanted it to fall down. The chaos of things falling down while he teaches is his schtick to add some levity and keep it interesting.
Did you know that the ratio of the side lengths to the length of a line connecting any two vertices that don’t share a side on a regular pentagon is the golden ratio?
Yeah the golden ratio is all over pentagons. This episode didn’t have time to go into the geometric traits, but we’ll look at that in a future episode someday
Oh damn, that's pretty useful actually. I came across a conspiracy theorist claiming that mathematicians are lying about the value of pi, that it isn't transcendental, but exactly equal to 4/√ϕ. Since you can calculate upper and lower bounds for the value of pi by constructing polygons around a circle, I bet using pentagons could produce a very elegant and rigorous way to disprove this crackpot theory.
Can you do a video on the Reimann hypothesis and the trillions of zeros. All the videos I watch leave me missing something I bet you could delineate in a down to earth way I could grasp better.
You remind me of myself, but you are about twice as smart and a whole lot younger. I have said for years that I see the universe as made up of numbers. They are everywhere. You have done a masterful job here with the Golden Ratio.
Woah, this is so cool! Love how you stumbled into the golden ratio so early :D Also, what happens if you plug in the golden ratio into the fibonnaci sequence?
Hey, Domotro, can you do a video on how many of the Golden Ratio exponents and Fibonacci Numbers are primes? And is there something to that, or is it just an interesting artifact??
Maybe! I do have a private mailbox address where people have sent clocks and stuff (listed in description) although I’m not sure how easy an easel would be to mail haha
why stop at reals? i bet you could make a "fibonacci-esque" sequence of complex numbers, and ratios of successive terms would still approach the golden ratio (at least in most cases)
I kind of wonder what it would look like to graph only the fractional part of powers of the golden ratio over the domain of all positive real numbers, but do it in such a way where points above 0.5 will be graphed below 0 (so have 1 subtracted) and points below 0.5 remain where they are. The result would be an oscillating curve that continuously approaches 0 forever. Obviously it gets close to 0 quite quickly, but is there anything interesting we can take from the oscillations? More interestingly, where would it appear the "zeroes" are in this graph if you were to look at every point it crosses the x axis? I might actually just make a python program to do this, I'll come back and share results if I do.
I did actually find that some people have already graphed this, which you can find pretty quick by searching "graph of golden ratio powers" online. I'll have to see if anyone has discussed the zeroes yet though.
Is the golden ratio related to the difference between powers? meaning the pattern 1²=1 1+3=2² 2²+5=3² 3²+7=4² as applied to powers of ³ or powers of ⁴ and so on
@@landsgevaer thank you so much, I've been trying to write an equation to calculate the difference between consecutive powers based on previous powers on and off for the past couple years, it's surprisingly a pain in the a**, especially with a highschool level maths skill 😅😅 if such an equation already exists no spoilers please
@@landsgevaer Is that the "2d" Fibonacci sequence thing? I had a feeling it was related to that someway, I guess there's a reason those numbers are so important 😅
Infinity hides almost everywhere. The numbers that don't are the vast exceptions. The vast majority of numbers aren't even calculable or definable properly. (Not that infinity hides. God hides.)
The finite decimal of golden ratio, when raised by powers will become a prime number… which is the a clue to the secret of my prime number generator… I’ve revealed to much 😮😮
How you accentuate your voice frustrates me I love ur videos and ur excitement but idk i find it kind of frustrating and hard to watch like i would watch a shit ton more of your videos and i love math But please just talk like ur normal self and take a joyous tone how u usually do but dont extend u syllables like how you do in this video it feels a bit pompous and talking downish I love the format of these videos and how irregular they are in terms of scenery and ur general personality (not saying your personality is irregular i mean i like it.)
i can't describe to you how repulsed i used to be by domotro's voice and manner of speaking. i kept watching despite that because it was just so interesting then i got used to it, and then i started genuinely enjoying his voice, so now i love it just give it a bit of time, i promise
8:05 missed opportunity to turn your 1s into phis by simply adding a circle, visually speaking, | + o = ϕ. makes me wonder if i make the ones in your ϕ^1s into phis as well, ϕ^ϕ, you better explain this domotoro upd: how many ways can you combine ϕ, -, and 1 to make .618...? ϕ-1 ϕ^-1 1/ϕ idk pls do a symbols video this ones driving me crazy
Thanks for watching! Check the description for some more notes, links, etc. :)
You may have heard it said that this is the closest thing to a "least rational" number. In a Numberphile video a few years ago, Ben Sparks said that the continued fraction being all 1s is why that is- the ratios of Fibonacci numbers are actually the closest approximations to it!
@@wyattstevens8574 Yeah I mentioned in the description of this video that a future episode will cover the cool irrationality-related traits of the golden ratio :)
Fib(n) = Round(Phi ^ n / SqrRoot(5)) ; Phi = GoldenRatio ~ 1.61803...
This was always my favorite as instead of tracking the last 2 values to add together, you can pluck it out by index with this spigot algorithm, which meant if you needed the n'th Fibonacci number, the old way you had to iterate n times to get to it, but this way you can calculate it immediately with no iteration.
Also, this is my favorite episode ever, clearly due to my affinity for this amazing value.
Highly underrated channel
if all mathemeticians were this handsome it would be so much easier learning maths
Dat brain too doe
🫵⏩🤪🏥
@@flickingbollocks5542 english?
Lol
-Also need higher pay.-
-For... costs.-
shit i read the comment as
"If all mathematicians were like this, it would be so much easier learning maths."
My mind is honestly blown right now. I love the way certain concepts seem to just weave themselves into another concept in so many different ways. Like, the golden ratio being in the Fibonacci sequence once is crazy enough, but _multiple times?!_ This is what I love about math, and I’m so glad there are passionate people like you out here making videos for people like me to enjoy and expand our knowledge of this wacky number world
I'm really liking the more clean-cut, organized, Combo Class. All the older episodes really triggered my cleaning OCD, and made it hard to concentrate.
The connections between phi and the square root of 5 are interesting. Obviously it's in one of the definitions of phi = (1+sqrt(5))/2, but also phi^3 = 2+sqrt(5); also, dividing a Lucas number by its corresponding Fibonacci number tends towards sqrt(5).
Sqrt(5) is in *every* non-zero integer power of phi.
I think 5 in general is interesting from a phi point of view.
For example every 5th number in the standard Fibonacci is divisible by 5. In the Lucas sequence no number is divisible by 5, but the Lucas Cassini identity, ((An)-1 * (An)+1) - ((An)^2 ), that is the product of two numbers either side of a number, minus the square of that middle number, equals + or -5.
In general 5 is hiding inside any what Domotro calls "Fibonacci-esque" sequence which contains no integer multiple of 5, such as his own "randomish" example at 11:02, 2 56 58 114 172 ... Note that 58*172 = 3020, so the Cassini identity for this sequence, + or -3020, is a multiple of 5. On the other hand any Fibonacciesque sequence which does go up in 5s, such as 2 57 59 116 175 ... has a Cassini identity of 3131, a non-multiple of 5.
I also found that one solution for 𝔁⁵ - 𝔁³ - 𝔁² - 𝔁¹ - 𝔁¹ - 𝔁⁰ = 0 is phi. Note the sequence of exponents is part of the Fibonacci sequence in reverse, beginning with 5. I don't think this works for any other number.
The continued fraction of sqrt(5) = [2; 4], which is twice the sqrt(2) = [1; 2]. Both these continued fractions belong to the very special class of continued fractions of sqrt(n^2+1). Next is sqrt(10). Aren't we beautifull!
Writing these as path informations where each digits turns to L if former is R and to L if former is R, in alternating manner looks like this (< for L and > for R for better visuals):
sqrt(2): > etc.
which looks even nicer when divided to rows
><
><
etc.
sqrt(5):
>>>>
@@chrisg3030 I always think about how we could justify using the decimal system to an alien. This might help
@@yasin_karaaslan It's quite fashionable these days to denigrate the decimal system in favor of the duodecimal for example, and I agree it's about time to reverse the trend. I'm not sure how I'd invoke the Fibonacci sequence for this, even though the 10th term using the convention of starting 1 0 1 1 ... is 55, which is also the sum of the first 10 natural numbers.
A more useful fact is that the first 4 natural numbers, 1 2 3 4, sum to 10. This means if you place a counter in one of the 4 quadrants of a cross hair grid, a giant + sign, say the top right and call it 1, move it to the bottom right and call it 2, bottom left 3, top left 4, then put another counter in the top right so now you have a representation for 5, and keep on moving and adding counters till all quadrants are occupied, you will then have a representation for 10.
Another possible stage in decimal counting history could have been the tic tac toe (noughts and crosses) grid. A counter in each successive space would represent 1 through 9, with an empty grid representing 0.
Domotro, I really can't praise this channel enough. Watching your channel now ignites the same spark of interest and passion for these topics that Numberphile inspired in me over a decade ago. I've fallen in and out of academics (I'm currently on an "extended hiatus" before finishing my engineering degree), but your videos remind me of why I loved math when I was younger and why I should love it again. Thanks.
You can even express powers of the golden ratio in terms of Fibonacci numbers
Golden Ratio^(n) = F(n) × Golden Ratio + F(n-1)
φ⁰ = 0φ + 1
φ¹ = 1φ + 0
φ² = 1φ + 1
φ³ = 2φ + 1
φ⁴ = 3φ + 2
φ⁵ = 5φ + 3
φ⁶ = 8φ + 5
φ⁷ = 13φ + 8
φ⁸ = 21φ + 13
φ⁹ = 34φ + 21
φ¹⁰ = 55φ + 34
And yes, the 0 case does work, the negative 1st Fibonacci number is also 1. You can work through the Fibonacci sequence backwords by thinking of what has to be added to the current first digit to equal the second digit. To get 1 from 0, you have to add 1 to it. Then to get 0 from 1, you have to add -1 to it, and so on.
I love how you only need 21 and 13 to make a good approximation of the golden ratio (21/13). Which is of course are part of the Fibonacci sequence (0 1 1 2 3 5 7 13 21 ...) Which impresses me even more because less than 10 elemnts and we're already almost there.
144/89
(1+ sqrt(5))/2 is my current favorite approximation of the golden ratio, but I hope they manage to get even closer
@@cherylchui4510 144=12^2. And sqrt(13) is the first continued fraction with greater than one odd number of repeating digits. Haven't yet figured out what might be the pattern of such numbers. Maybe, probably, somebody has, but dunno.
@@vampire_catgirl(1+sqrt(5))/2 is exactly the golden ratio.
you have the happiest-looking cats 🥺
Yeah they have a nice life. They get a lot of love/attention :)
It may be a little off topic but I'm so obsessed with this number I named my daughter after it(in a way) buy assigning numbers to letters (a:1, b:2, etc), for 1618 I got A,F,A,H and AFAH is her name.
OH wow I've only ever seen this channel in a popped out window at work I didn't realize how small it was. this is easily 100k views a video content. thank u for the knowledge, super interesting video
I'm very much looking forward to your new series on the golden ratio and the intricate connections between the Fibonacci and Lucas numbers. It will be interesting to see whether you explore polynomial expressions for F[2n], F[3n], F[4n], etc., and L[2n], L[3n], L[4n], etc., whose coefficients can be derived from Pascal's triangle and a simple variation on it.
i particularly like the setting, outside, Chaos happening, and the Cats! keep it going!
Oooh I learned some stuff. I didn't know the powers of phi got closer and closer to integers, much less the Lucas numbers. Very cool!
I would enjoy seeing an explananation in your style about why the golden ratio appears all over the place in pentagrams
A neat observation with the randomly chosen fibonacci-esque sequence is how quickly the first couple digits begin to resemble the traditional fibonacci sequence
When Brady from Numberphile generated a Fibonacci-esque sequence with 2 random numbers, it ended up being known as the "Brady numbers" and it's even in the OEIS. So we should call the sequence from this video the "Domotro numbers".
I like the idea, but let’s save the name “Domotro numbers” in case I discover something extra unique. If this sequence needs to be referenced for some reason it can be called Domotro’s Randomish Fibonacci-esque sequence
@@ComboClass Wise decision. We don't want you to confuse future generations of mathematicians like Euler did, by naming practically everything. :)
What about a number which added to its reciprocal equals that number squared? x + 1/x = x² ?
The one real solution (according to Wolfram Alpha anyway) is 1.4656. This number is the ratio constant in a sequence known as Narayana's Cows, just as the Golden Ratio is in the Fibonacci sequence. The difference is that every number is added not to the next but to the next-but-one. My own name for it is the Bovine Ratio, designated by the Greek character μ, pronounced "moo". The equation μ,+ 1/μ = μ² reads "Mummy moo plus baby moo equals moo squared".
I discovered the golden ratio at age 14 by looking for the number with the property 1/x = x-1. I also discovered that if you had a ravigneaux planetary gearset with a ring gear that is the golden ratio times larger than the sun gear (actually, both sun "gears" since there are two of them), you could make a four speed gearbox with ratios of 2.618, 1.618, 1.000, 0.618, and a reverse ratio of -1.618. (side note, I realize you can't make an irrational gear ratio with finite tooth count gears, but either approximate with large numbers or use discs with a high coefficient of friction)
You know, I've been watching this channel for awhile, but never thought to ask who the person holding the camera is.
This was filmed by my main camera guy Carlo. Some episodes have been filmed by other people too (who I name in the brief credits at the end of each episode)
@@ComboClass go Carlo :D
Before I even start this video, I have researched the golden ratio and it is already my favorite irrational number
I'm disappointed that you erased the 1s and replaced it with phis, instead of just drawing a 0 over the 1😁 8:11. I'm looking forward to the next episode on the golden ratio
Dude. I'm freaking out. I love your excitement regarding this and I feel as excited. I just figured out a thing I didn't know about Fibonacci numbers that relates to all this. Will share in discord
Bro you are something else
So, will you be going over the other metallic ratios?
In a future episode, sometime I will
thank you for the combo class
i use phi all the time; editing, sound, drawing. i didnt know about the phi^3 property maybe it works around to pi π
Lmao, I currently work at a bingo hall so the bingo ball cage was a bit of a trip, lmao. Since I've been working there tho I've realized theres a lot of complicated maths to do with bingo odds tho and so Ive been thinking a lot about combinatorics recently :) .
Two more facts about Phi:
The Greatest Common Denominator between any consecutive fibonacci numbers is always 1.
If you want to find the nth fibonacci number without calculating all previous numbers, the expression is ((phi)^n-(1/(phi))^n)/(sqrt(5))
Despite the equation being almost entirely made up of irrational numbers, an integer input will always produce an integer output.
One thing you didn't mention is that since a mile is ~1.61803 kilometers, you can use the Fibonacci sequence to approximate length conversions.
It’s 1.60934 but yeah that’s close enough to be useful for approximations! Neat!
@d3j4v00 I knew it wasn't exactly phi, that's why I put the tilde. Tildes mean "approximately".
@@nicholasweaver2374 Dude! I totally missed the ~
@@d3j4v00 It's okay.
That was great! .. reminds me of a numberphile with Ben Sparks who asserted that the golden ratio was "the most" irrational number since no numbers in the continued fraction are large.. in fact all ones , so it sorta threads that needle between rational numbers with the most room to spare on each side. There was a cool illustration with a graphic simulating a flower turning a certain amount each time it places another seed.. cool to see some other irrational numbers weren't so great because they're well approximated early on by a ratio (like pi had 3 spiral arms with wasted space ... eventually 22 spiral arms, etc.)
Yeah I love that topic. In the description of this vid I mentioned that a future episode will cover special traits the golden ratio has regarding irrationally
@Combo Class sweet!!.. you have at least 1 guaranteed view here brother : )
Also that was weirdly less chaotic than most episodes, love it still tho
I don’t think I will ever forget those identities after this enjoyable explanation. Now, can someone explain the Fibonacci spirals I see on memes?
Interesting topic.
This is a genius channel! Hopefully you get tons more views.
Amazing! … was just reading today about quadratic residue congruences for Fibonacci and Lucas numbers … would love if u could do an episode about quadratic reciprocity
Every time I think I've heard everything there is to know about the golden ratio, someone comes along and multiplies the amount I know by the golden ratio.
Here's comment 101! Love the clock decor, urban grunge and shameless use of a calculator. So refreshing after sanitized perfection videos.
All Fibonacci-esque sequence is basically the same, meaning once you find a formula for one, you can easily have the formula of any other.
0, 1, 1, 2, 3, ..., f(n)=f(n-1)+f(n-2)
0, u, u, 2u, 3u, ..., uf(n)
v, u+v, u+2v, 2u+3v, 3u+5v, ..., uf(n)+vf(n+1)
a, b, a+b, a+2b, 2a+3b, 3a+5b, ..., (b-a)f(n)+af(n+1) = af(n-1)+bf(n)
I did not expect that connection between fibonacci sequenceish and the golden ratio! Amazing lesson, goosebumps.
Your Channel is seriously underrated 😢
This channel is only a little more than 1 year old so I’m happy with its growth so far. Hopefully will continue to grow! Also my other “bonus” Domotro channel actually has more subscribers just because of all the shorts I’ve posted there
Same case here 😂 remember playing with the calc 😅
Great video! You might try getting an easel from hobby lobby or something. Would help with the whiteboards.
You must be a first time viewer. He wanted it to fall down. The chaos of things falling down while he teaches is his schtick to add some levity and keep it interesting.
You can also construct a cute identity for the golden ratio if you calculate its geometric series :)
Did you know that the ratio of the side lengths to the length of a line connecting any two vertices that don’t share a side on a regular pentagon is the golden ratio?
Yeah the golden ratio is all over pentagons. This episode didn’t have time to go into the geometric traits, but we’ll look at that in a future episode someday
Oh damn, that's pretty useful actually. I came across a conspiracy theorist claiming that mathematicians are lying about the value of pi, that it isn't transcendental, but exactly equal to 4/√ϕ. Since you can calculate upper and lower bounds for the value of pi by constructing polygons around a circle, I bet using pentagons could produce a very elegant and rigorous way to disprove this crackpot theory.
You are a madman. I Love it
Can you do a video on the Reimann hypothesis and the trillions of zeros. All the videos I watch leave me missing something I bet you could delineate in a down to earth way I could grasp better.
Yeah I am planning an episode about the Riemann hypothesis to come out at some point in the future
You remind me of myself, but you are about twice as smart and a whole lot younger. I have said for years that I see the universe as made up of numbers. They are everywhere. You have done a masterful job here with the Golden Ratio.
For the Fibonacci esce sequence what if it's a log as the first 2
Like ln1 ln2 » ln2 » ln 4 etc
Does it still hold?
Woah, this is so cool! Love how you stumbled into the golden ratio so early :D
Also, what happens if you plug in the golden ratio into the fibonnaci sequence?
If you make a Fibonacci esque sequence starting from 1 then the golden ratio, the rest of the sequence would be the further powers of the golden ratio
Golden ratio cubed has a tail of sqrt5. But generally golden ratio + it's reciprocal = sqrt5
Hey, Domotro, can you do a video on how many of the Golden Ratio exponents and Fibonacci Numbers are primes? And is there something to that, or is it just an interesting artifact??
I am not aware of anything special there.
See en.m.wikipedia.org/wiki/Fibonacci_prime
Wow. I thought the golden ratio was really cool. Little did I know it's actually really really cool.
(x^1)+1 = x^(1+1)
Solve for x. The bracket shift rule. The structure is more apparent using the caret ^ rather than a superscript for the exponent.
Are there any interesting things about the Eisenstein integers that you know of?
If I sent you an EASEL, would you use it? btw, I really enjoy your shows!
Maybe! I do have a private mailbox address where people have sent clocks and stuff (listed in description) although I’m not sure how easy an easel would be to mail haha
Also, Have you ever discussed X^X? min value = 1/e... why e??? @@ComboClass
This dude's set be looking like one of those "find the object" mobile games...
why stop at reals? i bet you could make a "fibonacci-esque" sequence of complex numbers, and ratios of successive terms would still approach the golden ratio (at least in most cases)
Dude erased a one to write the golden ratio symbol thing (that contains a vertical line)
it was cool when I stumbled upon cos(72º) * 2 in high school
I kind of wonder what it would look like to graph only the fractional part of powers of the golden ratio over the domain of all positive real numbers, but do it in such a way where points above 0.5 will be graphed below 0 (so have 1 subtracted) and points below 0.5 remain where they are. The result would be an oscillating curve that continuously approaches 0 forever. Obviously it gets close to 0 quite quickly, but is there anything interesting we can take from the oscillations? More interestingly, where would it appear the "zeroes" are in this graph if you were to look at every point it crosses the x axis?
I might actually just make a python program to do this, I'll come back and share results if I do.
I did actually find that some people have already graphed this, which you can find pretty quick by searching "graph of golden ratio powers" online. I'll have to see if anyone has discussed the zeroes yet though.
Over the reals? Then it doesn't converge to zero at all.
Over the integers, it does, by a factor -1/phi every step
You sound like Kevin from vsauce2 I can't stop hearing it
1 minute ago 💀
Maybe ~1.618 minutes now
@@ComboClass probably
The GR has quite a pretty continued fraction....
You missed one the diagonal of a regular unit pentagon is equal to the golden ratio.
Is the golden ratio related to the difference between powers? meaning the pattern 1²=1 1+3=2² 2²+5=3² 3²+7=4² as applied to powers of ³ or powers of ⁴ and so on
Not particularly.
The difference between n-th powers go like an (n-1)-th power, roughly.
@@landsgevaer thank you so much, I've been trying to write an equation to calculate the difference between consecutive powers based on previous powers on and off for the past couple years, it's surprisingly a pain in the a**, especially with a highschool level maths skill 😅😅 if such an equation already exists no spoilers please
@@christiannersinger7529 It exists.
Not a spoiler but a hint
..
..
..
You know Pascal's triangle?
@@landsgevaer Is that the "2d" Fibonacci sequence thing? I had a feeling it was related to that someway, I guess there's a reason those numbers are so important 😅
@@christiannersinger7529 It is 2D, but not Fibonacci. AFAIK
What if it stsrts with 1 and phi
1:49 kitty
3:35 2 kitty!
If you look closely at the outro shot you can see all 3 cats :)
Wow that’s CRAZY
Me watching this: I wonder when the Lucas numbers will pop up
Me at 15:00 😁
I like the story about your life thanks
Omg we're nearly 2 minutes in and nothing has fallen off the table or caught on fire… what is happening?? 😅
Dayum, it took till 11:35!! 🤯
Sorry Domotro. I have a hard time focusing in what you say in that part of the video with that gorgeous cat at your side. 😻
wow that's so cool
13:20 -1/phi is a repelling fixed point of 1/(1+x) while phi is attracting
I didn't know it applies here too though
16:04 omg
Is… THAT A JOJO’S REFERENCE???
The Golden Ratio is where God hides Infinity.
I dunno man Pi is pretty dope too
Infinity hides almost everywhere. The numbers that don't are the vast exceptions. The vast majority of numbers aren't even calculable or definable properly.
(Not that infinity hides. God hides.)
2phi+1
I've always found it funny that a number called the "golden ratio" is irrational.
Metallic
x³ = x + 1
plastic
I'm disappointed that you erased the 1s and replaced it with phis, instead of just drawing a 0 over the 1😁
It’s not impossible to calculate a finite decimal value …..
The finite decimal of golden ratio, when raised by powers will become a prime number… which is the a clue to the secret of my prime number generator…
I’ve revealed to much 😮😮
The numbers have been greatly exaggerated
Which numbers? Unless you’re just joking about how I said near the beginning that some other aspects of the golden ratio are exaggerated
@@ComboClass I think that was the joke yea 😅
@@ComboClass it was also a reference to "reports of my death have been greatly exaggerated" haha. Thanks for the vid!
yay
phith
x³=x²+x+1
???
kill a a a nutshell
It is irrational and rational….
!
How you accentuate your voice frustrates me
I love ur videos and ur excitement but idk i find it kind of frustrating and hard to watch like i would watch a shit ton more of your videos and i love math
But please just talk like ur normal self and take a joyous tone how u usually do but dont extend u syllables like how you do in this video it feels a bit pompous and talking downish
I love the format of these videos and how irregular they are in terms of scenery and ur general personality (not saying your personality is irregular i mean i like it.)
i can't describe to you how repulsed i used to be by domotro's voice and manner of speaking. i kept watching despite that because it was just so interesting
then i got used to it, and then i started genuinely enjoying his voice, so now i love it
just give it a bit of time, i promise
also try putting the video on 1.25x speed, maybe it will make it more pleasant to you
@@ezhanyan thank you for your advice I will do it from now on
Phi to the n-th power equals the n plus one-th fibonacci number times phi plus the n-th fibonacci number
φ^n=φ*fib(n+1)+fib(n)
you get this if you extend the φ^n=φ^(n-1)+φ^(n-2) identity
8:05 missed opportunity to turn your 1s into phis by simply adding a circle, visually speaking, | + o = ϕ. makes me wonder if i make the ones in your ϕ^1s into phis as well, ϕ^ϕ, you better explain this domotoro
upd: how many ways can you combine ϕ, -, and 1 to make .618...?
ϕ-1
ϕ^-1
1/ϕ
idk pls do a symbols video this ones driving me crazy
Haha good point, that would have been smoother to write the symbols that way. And yeah that’s a fun question to think about
ϕ^(1 +1 ) = (ϕ^1) + 1
The bracket shift rule.