At 7:23 you made a small mistake because the very next line is not an exact statement, but an approximation, which is only true for n going to infinity.
+PlopKonijn I did. He mentions it but he doesn't acknowledge that this video is kind of pointless because he wants to prove his point by using the same trick (approximating) he did last time - which he was criticized for by the Reddit user.
+anononomous But hey, at least it would be a slightly different reading of maths as they exist in the real world, so that's a step up from *cough* some things *cough* .
anononomous: ...kinda like the conflict between the Palestinian Liberation Front and the Liberation Front Of Palestine and the Front For The Liberation Of Palestine?
I've never laughed so hard at a Numberphile video. As soon as I realized Matt was circling back to his favored Lucas sequence I lost it. That delivery was perfect Matt!
@Teun van Diedenhoven That is true, if you consider 0 to be Fibonacci number #1; rather, than Fibonacci number #0. Matt was considering the fibo numbers to start from 1, 1,…, in which case, both 1 and 5 would meet the criteria; although, either way, 1 occupies 2 positions (#0 & #1, or #1 & #2).
@@goutamboppana961 The golden ratio doesn't equal exactly the next Fib. number divided by the current. The division between consecutive Fibonacci numbers is an approximation of the golden ratio, and if you assume it's exactly the same, you get the result Mr. Parker is showing. There's the sneaky rounding!
As a math student named Lucas, I cannot describe how amazing it feels to have the great Matt Parker describe why Lucas numbers are better than Fibonacci numbers...
I feel like Lucas numbers versus Fibonacci numbers debate is kind of like pi versus tau...both of some advantages but they're closely related so it doesn't really matter which one
Yup, reminded me of that debate as well, minus the little difference how Parker was on the popular side of the argument with Pi vs Tau (picking Pi's side) whereas here he's in the less popular side, fighting against the very common/very popular Fibonacci sequence and the Golden Ratio, lol. We've seen him tackle this topic before though too so the opinions he expressed here weren't too surprising given that us long time viewers already knew what to expect ;)
"Proxy Pylon" is actually the name of an opening gambit you can perform in the StarCraft/StarCraft 2 games. It's considered to be a 'cheap' tactic, so I'm glad you weren't beaten by it :D
Actually "proxy something" refers to basically any production facility (or a pylon) placed strategically outside your base to either conceal your plans or shorten the time needed for your units to reach the desired position. It can be used in a cheesy way to one-base someone into oblivion but these are also common during the middle and sometimes even late game. Proxy pylons especially.
Matt: 5 is the only fibonacci number equal to its position First fibonacci number: they ask you how you are, and you just have to say you're fine when you're not really fine, but you ...
Did anybody else feel a little thrill of anticipation when Matt said “let’s generalize it and call it a day”? Like, oh boy, can’t wait to see how he burns the internet back :D
I have used a spreadsheet to work out which fractions m/n best approximate to the golden ratio as n increases. For n=1, the closest approximation is 2/1. For n=2 it is 3/2. For n=3 it is 5/3. For n=4 there is no approximation better than 5/3. For n=5 the closest approximation is 8/5. The next n which produces a closer approximation is n=8, for which 13/8 becomes the best approximation to the golden ratio. After that better approximations are achieved by is 21/13 and then 34/21. I didn't continue the spreadsheet any further. It is the Fibonacci numbers which are clearly providing the best approximations. 34/21 is accurate to within 0.0010 whereas (for example) 47/29 is out by 0.0026
The generalized sequence also works in reverse, to find Fibonacci numbers with indexes zero or lower. Before seeing this, I never thought to go in the other direction. Pretty neat!
1:05 "In fact, any sequence where you start with two numbers and then add them together next one and repeat, always approaches the golden ratio." 0, 0 -> 0, 0, 0, 0, 0, 0, ...
1. Matt thinks Lucas numbers are better than Fibonacci numbers 2. Lucas numbers are better because otherwise you need to split it into 2 sets of fibonacci numbers to accomplish the same thing 3. You need two sets of pi to get Tau 4. Tau must be better than Pi because otherwise you need to split it into 2pi to accomplish the same thing 5. Matt must think that Tau is better than Pi.
Love the video! Two quick corrections: 1:08 Not any two starting values will generate a "fibonacci sequence," for you could start with 0 and 0. 6:52 5 is not the only Fibonacci Number which is the same as its position. The other is of course 1!
This is a fun little back and forth. And in the end... it just turns out to be one of those things not worth arguing about, because EVERYONE IS RIGHT. We all tend to have a preference for things we are most familiar with - we get to stay more in our "comfort zone." Doesn't make us "right" and someone else "wrong."
I am not convinced I am on zeproxypylon's side on this one that rounding step is just too ugly for me. p.s.: this new argument is almost like a parker square.
No, you can't assume that Fn*phi = Fn+1. That would be rounding since the ratio between Fn and Fn+1 only aproaches phi. You can only get the lucas numbers by doing some kind of rounding. Edit: Wait, you talked about it
Kind of a null point since you can just generalise the Lucas numbers back into the Fibonacci numbers; personally, I'm with zeproxypylon on this since he was actually able to get Matt Parker to admit he was wrong (sort of).
You can take any of the sequences and add the surrounding digits to forma new one. For example, the Lucas numbers, using the same formula, generate 5,5,10,15,25,40 and so on, which then can generate 15,20,35,55,90
7:23 - Parker Generalisation. I don't believe that F(n) * Phi = F(n+1), because as already explained in the video, the golden ratio is what the Fibonacci numbers tend to as a ratio between them, so does not yield perfect results prior to infinity, which is quite a lot of numbers, to say the least, so will not be a correct generalisation due to inaccuracy. Let's take the 5th number. The 5th Fibonacci number is 5. Phi ^ 5 = 11.0901699.... Using the Parker Generalisation: F(n+1) + F(n-1), we get 3 + 8 = 11. Of course, 11 ≠ 11.0901699... So we have proven this to be wrong. Edit: nevermind... Didn't watch till 10:00.
You can represent the Fibonacci Numbers as F(x)=(φ^x-cos(πx)φ^-x)/√(5) And the Lucas Numbers as L(x)=φ^x+cos(πx)φ^-x And, in general, for any sequence with initial values S(1) and S(2), and with the same recurrence relation as Lucas and Fibonacci, we can write our sequence as S(x)=A*φ^(x-1)-Bcos(πx)*φ^(1-x) with A and B such that A=(S(1)(φ-1)+S(2))/√(5) B=(S(1)φ-S(2))/√(5) (Also, for some reason, the first two formulas for Lucas and Fibonacci share some sort of symmetry that reminds me of the relationship between cosines and sines, and I can see what he's saying about how they're sort of tied together)
You can get around the rounding another way. If you let PHI be (1+root5)/2 and phi be (1-root5)/2 ie. the two roots of x^2=x+1, then the nth Lucas number, Ln = PHI^n + phi^n.
But if you do the same with Lucas numbers you get Fibbonacci numbers. Well, multiplied by 5, but still. So Fibbonacci numbers are based on Lucas numbers, wich are based on Fibbonacci numbers wich are ba... ~[1 infinity later]~ In fact, in similar way it's possbile to construct any Fibbonacci sequence from any other you (just need to multiply these numbers by some factors) for example to make the third sequence (3,1,4,5... (I forgot the name)) from Fibbonacci you need to take a Fibbonaci number, multiply by 5, then add the prevoius one multiplied by -2
Circular reasoning. You've assumed that the Fibbonnaci numbers have been pre-defined in order to define the Lucas numbers. You can just as easily do the reverse, and define the Fibbonnaci numbers in terms of the Lucas numbers. But in my view, what makes the Fibbonnaci numbers more basic, is that they use the recursion that both sequences use, but with the simplest non-trivial starter pair: (0, 1). Every sequence a(n) that uses the Fibbonnaci recursion, can be written as a linear function of F(n) and F(n-1). And in particular, every integer sequence a(n) that uses that recursion, can be written as an integer linear function of F(n). Fred
but there is a small thing when matt multiply golden ratio with nth fibonacci no. matt did rounding as the ratio of consecutive fibonacci no. approach phi
i'd have to disagree on that because the earliest number aren't of much interest if you want a precise value of the golden number. We are talking about converging speed and we can see that in fact this series converge faster to the golden number than the fibonachi one. thus you'd need less calculus to approach the rounded value to the n-th decimal to get it hence its usefulness. edit: though a bit of mathematic rigor would be welcomed as his demonstrations reminds me of how i did maths in HS...
So the biggest take away from this is closing your eyes and rounding in prayer will give you any set of numbers you like to fit any argument! See I can maths two!
But that is also true about Fibonacci: 1, 1, 2 . . . 3rd term divided by the second is 2/1 = 2, not phi. They approach the actual golden ratio in the limit as the number of terms approaches infinity, because only then will the ratio between two (extremely large) integers begin to approach an irrational number.
Sebastian Elytron That has to come with a sequence known as Narayana's Cows (OEIS A000930) with a recurrence Cn = Cn-1 + Cn-3. The ratio between successive terms is approx. 1.4656. We could call that the Beefy ratio designated by the Greek character Moo. Moo^2 - Moo = 1/Moo
You know, in the earlier Lucas Numbers video, the rounding seemed a bit weird to me too, but when you point out here that it's equivalent to pretending that the much-vaunted Golden Ratio of the Fibonacci numbers is actually the exact ratio and not just the limit that it tends to, that makes that earlier video feel much more intuitive.
As another thing in defence of the Fibonacci sequence being tied to the Golden Ratio, if you look at the continued fraction of the Golden Ratio and look at its convergents, then they are F_{n+1}/F_n where F_n is the nth Fibonacci number. Which means that, for example, 21/13 is the ratio of two consecutive Fib. numbers and you can't find a rational number closer to the Golden Ratio with a denominator less than or equal to 13. So F_{n+1}/F_n is the closest rational number to \phi relative to F_n (relative to F_n meaning that there doesn't exist an p/q closer to \phi if b is less than or equal to F_n). So, in that sense the Fibonacci sequence is the fastest converging sequence to \phi.
![Felix "Unfair" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/Oswujxm2Ag0/mqdefault.jpg)
Part 2 is at: ruclips.net/video/z1THaBtc5RE/видео.html
Check out some Numberphile T-Shirts and other stuff: teespring.com/stores/numberphile
Numberphile is this and old video? Matt has shaved his head on his channel
At 7:23 you made a small mistake because the very next line is not an exact statement, but an approximation, which is only true for n going to infinity.
@@SaborSalek watch the whole video before commenting please
+PlopKonijn
I did. He mentions it but he doesn't acknowledge that this video is kind of pointless because he wants to prove his point by using the same trick (approximating) he did last time - which he was criticized for by the Reddit user.
@@SaborSalek no he does acknowledge it. He talks about the rounding error.
Matt has two expressions: pleased with himself, and displeased with someone else
@Dr. M. H. hahaha xD
*laughing from US
666 likes ooooooh spooky
@@ryanmunn4134 0 likes spooky
@@monasimp87 000000h spooky 👻
@@ryanmunn4134 1200 likes ooooooh spooky
The video is 11:23 long, what an ingenious "coincidence"!
You must be on mobile... It adds another second for no reason.. Sorry to tell the video is actually 11:22 long...
Or is it?
Vsauce music plays
@@fdnt7_ Vsauce, Michael here. Is time theft a thing?!
It rounds it up....
"...Let's do what we do to celebrate things in mathematics, let's try to generalize them"
WOOOOO PARTY!!!
When he said that I paused to check whether someone already commented about it :-D
Celebrating a job well done by taking it into overtime. Proof that you love your work.
"I should give him directions to the nearest... maths... department-what?"
This is why I love Matt
I am actually dying of laughter right now and in tears typing because of this edit
The Lucas numbers should be classified as a Parker Sequence due to their almost correctness.
THIS is the real burn. Well played.
Exactly 🎯! 👍🏻
The Golden Trilogy: an epic saga on the war between the Lucasians and the Fibbonaccis.
Having a war over a slightly different reading of what is effectively the same thing... Nah, would never happen...
REEEEE
+anononomous But hey, at least it would be a slightly different reading of maths as they exist in the real world, so that's a step up from *cough* some things *cough* .
anononomous: ...kinda like the conflict between the Palestinian Liberation Front and the Liberation Front Of Palestine and the Front For The Liberation Of Palestine?
@@shruggzdastr8-facedclown you mean kinda like the conflict between the people's front of judea and the judean people's front?
I've never laughed so hard at a Numberphile video. As soon as I realized Matt was circling back to his favored Lucas sequence I lost it. That delivery was perfect Matt!
That Parker Square at 6:05
Sneaky bastards
I think that's the same one from a different angle…
Soon after, the link to merch appeared.
@2:40
The op was referring to the one that flashed onto the picture on the wall at 6:05, not the one on the desk.
”5 is the only Fibonacci number that’s equal to its position.”
1: ”Am I a joke to you?”
@Teun van Diedenhoven That is true, if you consider 0 to be Fibonacci number #1; rather, than Fibonacci number #0. Matt was considering the fibo numbers to start from 1, 1,…, in which case, both 1 and 5 would meet the criteria; although, either way, 1 occupies 2 positions (#0 & #1, or #1 & #2).
1's the Schrödinger's Fibonacci number; literally in the right place and the wrong place at once.
Not to mention 0, the 0th Fibonacci number.
@Teun van Diedenhoven they do start with 0, but they start with the 0th number in the sequence, not the 1st.
@Teun van Diedenhoven 0 and 1 do. You just listed 2 counter examples in your comment lol.
7:23 I am calling Matt out on this hidden and sneaky rounding.
YES I thought the same thing hahah
explain plz i am curious
@@goutamboppana961 The golden ratio doesn't equal exactly the next Fib. number divided by the current. The division between consecutive Fibonacci numbers is an approximation of the golden ratio, and if you assume it's exactly the same, you get the result Mr. Parker is showing. There's the sneaky rounding!
Matt admits his hidden and sneaky rounding at 9:51
What an exciting time to be alive
How are you verified?
Why are you verified?
Where are you verified?
As a math student named Lucas, I cannot describe how amazing it feels to have the great Matt Parker describe why Lucas numbers are better than Fibonacci numbers...
I am quite happy that Matt did another Numberphile. He has a very nice presentation.
I feel like Lucas numbers versus Fibonacci numbers debate is kind of like pi versus tau...both of some advantages but they're closely related so it doesn't really matter which one
*pi vs tau
@@harshsrivastava9570 oops typo thanks
Yup, reminded me of that debate as well, minus the little difference how Parker was on the popular side of the argument with Pi vs Tau (picking Pi's side) whereas here he's in the less popular side, fighting against the very common/very popular Fibonacci sequence and the Golden Ratio, lol. We've seen him tackle this topic before though too so the opinions he expressed here weren't too surprising given that us long time viewers already knew what to expect ;)
“so it doesn’t really matter which one”
...except pi is superior.
I used to think π was better but then I did complex analysis and the amount of times you have to write 2π is annoying
"Proxy Pylon" is actually the name of an opening gambit you can perform in the StarCraft/StarCraft 2 games. It's considered to be a 'cheap' tactic, so I'm glad you weren't beaten by it :D
and "ze" probably means "the".. and we need additional pylons!
I am so glad someone caught that! My life for Aiur!
Actually "proxy something" refers to basically any production facility (or a pylon) placed strategically outside your base to either conceal your plans or shorten the time needed for your units to reach the desired position. It can be used in a cheesy way to one-base someone into oblivion but these are also common during the middle and sometimes even late game. Proxy pylons especially.
I was waiting with bated breath for someone who knew more SCII stuff to give me the deep dive on the strats like this. Thanks :P
The meaning of his account was the only part of the video I could understand.
6:50 "5 is the only Fibonacci number which is equal to its position"... what about 1?
1,1 so 1's position is first AND second so it's position is 1.5 and it's approximately 2
"so it's position is 1.5 and it's approximately 2"
Wow, hold your horses! I was here to do maths, not physics :P
@@amxx if u watch favremysabre when u say horses the horse that talks is Lucas
So its like a joke
That is a trivial case.
Matt: 5 is the only fibonacci number equal to its position
First fibonacci number: they ask you how you are, and you just have to say you're fine when you're not really fine, but you ...
I guess that’s, what we call: a ”Parker Fun Fact” 😅.
No that was an unexpected turn of events. Always finding new ways to never admitting defeat. 👏😂
Matt, you are a true man's man! 👍
A true math's man.
Oh, Matt is admitting he is wrong... wait! He's turned it around! He is right again!! Hooray!!! (I'm a fan of Matt Parker, in case you didn't notice.)
That "plot twist" is so beautiful.
Just a abusing an equal sign here or there
Yeah, irrational number equals integer. Hrmmmm.
And then he was wrong again
Parker square...
Did anybody else feel a little thrill of anticipation when Matt said “let’s generalize it and call it a day”? Like, oh boy, can’t wait to see how he burns the internet back :D
I did 😅.
I remember this back when it was posted on his subreddit over an year ago, it took you guys a long time to get around to it.
I don't know which is better, Matt's epic comeback or the fact that this video is exactly 11:23 minutes long...
"It turns into a bit of a philosophical discussion about the square root of five" is a phrase you just KNOW involves Matt Parker somehow.
Matts hair grew back!
The sequence of Matts head tending towards a sphere is not convergent, it turns out.
Only some of it :P
Some of it at least ;)
What happened to it in the first place? I seem yo be living under a rock
This is probably an earlier recording before he shaved it.
We he said F_n*phi= F_n+1, he was rounding. That's only true as n tends to infinity.
Exactly!!!! It wasn't a good burn
Yeah, good that other people also caught it. We should upvote all the comments that mention this so that Matt and Brady realize it.
Theo_Caro YES ! Oh my god ! I was like WHAT IN THE WORLD IS HE DOING ??
He said that in the end ^^
He even said that himself. But at least, there's a comment for the system. gj
Why the Fibonacci numbers are better: if you stop the continued fraction of the golden ratio at finite points, you get ratios of Fibonacci numbers
Both approaches are awesome. Mind blown.
There's a lot of reaching in both arguments methinks 😂
FutureNow Hey when are you going to start making more videos?
notKARTHIK. Hey, so my upload schedule right now is roughly once per month so there will be a new video by this weekend.
welcome to arguments in the internet
Well technically you reaching tending towards infinity and then it works perfectly yeah.
Maybe he just wants to be Golden ratio'd.
I love it that the moment Matt said he always admits when he's wrong, a link popped up for Parker Square merchandise :D Well played.
How dare you admit that you were wrong without comparing your oponent to Hitler , this is not how internet arguments are supposed to work!
I love that when Parker admits he is wrong @6:08 the card pops up saying: *want to buy some Parkersquare merchandise?*
Love it!
I have used a spreadsheet to work out which fractions m/n best approximate to the golden ratio as n increases.
For n=1, the closest approximation is 2/1. For n=2 it is 3/2. For n=3 it is 5/3. For n=4 there is no approximation better than 5/3. For n=5 the closest approximation is 8/5. The next n which produces a closer approximation is n=8, for which 13/8 becomes the best approximation to the golden ratio. After that better approximations are achieved by is 21/13 and then 34/21.
I didn't continue the spreadsheet any further. It is the Fibonacci numbers which are clearly providing the best approximations. 34/21 is accurate to within 0.0010 whereas (for example) 47/29 is out by 0.0026
Yes, this fact is actually obvious because of the continued fraction of the golden ratio.
I was waiting for it, Matt did not disappoint.
Today getting video from 3Blue1Brown and Numberphile😍😍😍
That is nice
The generalized sequence also works in reverse, to find Fibonacci numbers with indexes zero or lower. Before seeing this, I never thought to go in the other direction. Pretty neat!
Looks like he was more of an Artosis Pylon.
Damn I love this channel. Fascinating content as always.
Golden Age of Meme
1:05 "In fact, any sequence where you start with two numbers and then add them together next one and repeat, always approaches the golden ratio."
0, 0 -> 0, 0, 0, 0, 0, 0, ...
They're good sequences, Brent.
6:05 love the "That's a classic Parker Square move" in the upper right!
1. Matt thinks Lucas numbers are better than Fibonacci numbers
2. Lucas numbers are better because otherwise you need to split it into 2 sets of fibonacci numbers to accomplish the same thing
3. You need two sets of pi to get Tau
4. Tau must be better than Pi because otherwise you need to split it into 2pi to accomplish the same thing
5. Matt must think that Tau is better than Pi.
“A bit fuzzy and almosty” - so it was the Parker Square basically.
9:24 classic parker joke
Love the video! Two quick corrections:
1:08 Not any two starting values will generate a "fibonacci sequence," for you could start with 0 and 0.
6:52 5 is not the only Fibonacci Number which is the same as its position. The other is of course 1!
BUUURRRRNNN!
_Fun and informative video; _*_thanks_*_ for doing this_ 👍
This is like a mathematical rap battle
Backwards Fibonacci
5, 3, 2, 1, 1, 0, 1, -1, 2, -3, 5
Backwards Lucas
11, 7, 4, 3, 1, 2, -1, 3, -4, 7, -11
EDIT: Whoa, what's this? A second like bomb?
Palindrome sequence; I like that!
Interesting
backwards sequence
5x, 4x, 3x, 2x, x, 0, -x, -2x, -3x, -4x, -5x
SPOOOOOKKKKKYYYYYY COIIINNCCCIIIDDDEENNNSSSCCCSSSCCSCSCCSCSCCSCSSCSSSSSSSSSSSSSSS
Backwards Fibonacci is actually
5, 3, 2, 1, 1, 0, 0, 0...
+[Slight Lokii]
1 - 0 = 1 though, and not 0.
This is a fun little back and forth. And in the end... it just turns out to be one of those things not worth arguing about, because EVERYONE IS RIGHT. We all tend to have a preference for things we are most familiar with - we get to stay more in our "comfort zone." Doesn't make us "right" and someone else "wrong."
So doesn't that mean the Fibonacci numbers generate the Lucas numbers which makes them (the Fibonacci numbers) more fundamental?
Yes, but I don't think Parker likes highlighting that little aspect... ;)
That depends on the point of view. You can also turn this statement around and say the opposite.
From my limited observations, adding the Lucas Numbers in the same way gives you the fibonacci sequence multiplied by 5.
I can't imagine Numberphile without the markers and brown paper, but by God the sound it makes is like nails on a chalkboard for me!!
I am not convinced I am on zeproxypylon's side on this one that rounding step is just too ugly for me.
p.s.: this new argument is almost like a parker square.
GOSH I LOVE THIS CHANNEL AND I LOVE MATH
No, you can't assume that Fn*phi = Fn+1. That would be rounding since the ratio between Fn and Fn+1 only aproaches phi. You can only get the lucas numbers by doing some kind of rounding.
Edit: Wait, you talked about it
If sounds to me like the fibonacci sequence is just Lucas numbers with extra steps
Kind of a null point since you can just generalise the Lucas numbers back into the Fibonacci numbers; personally, I'm with zeproxypylon on this since he was actually able to get Matt Parker to admit he was wrong (sort of).
You can take any of the sequences and add the surrounding digits to forma new one. For example, the Lucas numbers, using the same formula, generate 5,5,10,15,25,40 and so on, which then can generate 15,20,35,55,90
Also, if you work out the simple formula, you get: a,a+b,2a+b,3a+2b,5a+3b,8a+5b, and so on, giving you two more sets of Fibonacci numbers
The Parker Square merch card at 6:00 when he admitted he was wrong was hysterical.
6:15, “Let’s do what we do to celebrate in mathematics, we try to generalise them”.
You know Matt’s got something up his sleeve when he says this 😂😂
Aw man I love this channel
7:23 - Parker Generalisation. I don't believe that F(n) * Phi = F(n+1), because as already explained in the video, the golden ratio is what the Fibonacci numbers tend to as a ratio between them, so does not yield perfect results prior to infinity, which is quite a lot of numbers, to say the least, so will not be a correct generalisation due to inaccuracy.
Let's take the 5th number. The 5th Fibonacci number is 5. Phi ^ 5 = 11.0901699.... Using the Parker Generalisation: F(n+1) + F(n-1), we get 3 + 8 = 11. Of course, 11 ≠ 11.0901699... So we have proven this to be wrong.
Edit: nevermind... Didn't watch till 10:00.
You can represent the Fibonacci Numbers as
F(x)=(φ^x-cos(πx)φ^-x)/√(5)
And the Lucas Numbers as
L(x)=φ^x+cos(πx)φ^-x
And, in general, for any sequence with initial values S(1) and S(2), and with the same recurrence relation as Lucas and Fibonacci, we can write our sequence as
S(x)=A*φ^(x-1)-Bcos(πx)*φ^(1-x)
with A and B such that
A=(S(1)(φ-1)+S(2))/√(5)
B=(S(1)φ-S(2))/√(5)
(Also, for some reason, the first two formulas for Lucas and Fibonacci share some sort of symmetry that reminds me of the relationship between cosines and sines, and I can see what he's saying about how they're sort of tied together)
L_n = phi^n + (1 - phi)^n
the true "no rounding" version
'Im going to replace an approximation (the rounding) by another approximation". Isn't that an auto-burn?
7:30 Fn + φ = F(n+1)? That doesn't sound right.
You can get around the rounding another way. If you let PHI be (1+root5)/2 and phi be (1-root5)/2 ie. the two roots of x^2=x+1, then the nth Lucas number, Ln = PHI^n + phi^n.
"five is the only Fibonacci number that is equal to its position"
Correct me if I'm wrong, but doesn't it start with one?
those parker square popups are so much on point in all your videos
It's almost as if the Lucas Number are BASED on the Fibonacci Numbers!
It's actually the other way around
How so?
Harsh Srivastava Fibonacci published his number in Liber Abaci in 1202.
beirirangu CAPITALIZING words doesn’t make your ARGUMENT any better
But if you do the same with Lucas numbers you get Fibbonacci numbers. Well, multiplied by 5, but still.
So Fibbonacci numbers are based on Lucas numbers, wich are based on Fibbonacci numbers wich are ba...
~[1 infinity later]~
In fact, in similar way it's possbile to construct any Fibbonacci sequence from any other you (just need to multiply these numbers by some factors) for example to make the third sequence (3,1,4,5... (I forgot the name)) from Fibbonacci you need to take a Fibbonaci number, multiply by 5, then add the prevoius one multiplied by -2
Omg Matt. I think you summoned the evil know :-) Very nice Video. I love that "Burned with your own arguments"-discussions :-) Thanks
Unless you can get the Lucas numbers out of Pascal's Triangle more simply than the Fibonacci sequence, Fibonacci wins hands down.
9:03 I noticed that rounding 😈.
*EDIT:* 9:57 Exactly 👌🏻🎯!
If Lucas numbers are the Fn+1 and the Fn-1 together, then their origin is Fibbonnaci (himachandra). There is no debate.
Circular reasoning. You've assumed that the Fibbonnaci numbers have been pre-defined in order to define the Lucas numbers.
You can just as easily do the reverse, and define the Fibbonnaci numbers in terms of the Lucas numbers.
But in my view, what makes the Fibbonnaci numbers more basic, is that they use the recursion that both sequences use, but with the simplest non-trivial starter pair: (0, 1).
Every sequence a(n) that uses the Fibbonnaci recursion, can be written as a linear function of F(n) and F(n-1).
And in particular, every integer sequence a(n) that uses that recursion, can be written as an integer linear function of F(n).
Fred
but there is a small thing when matt multiply golden ratio with nth fibonacci no. matt did rounding as the ratio of consecutive fibonacci no. approach phi
10:34 classic parker phrase
Reflecting on ones own mistakes is a most beautiful thing.
Wait....... Why does matt has a full set of hair... Hmm suspucious (?)
Tuvi Its not the real Matt Parker. He’s more of a Parker Matt Parker.
Pre recorded and released now o.o I thought about this Tuvi
4:00 Well, surely 'not very precise' and 'rough and ready' are familiar terms for Matt 'Parker Square' Parker.
I still think that hidden rounding effort counts as cheating. zeproxypylon gets my vote
100% agree
Yep! He does exactly the same rounding by saying F_n*phi = F_(n+1). Zeproxypylon is correct
Right. Even worse when one tries to hide it: I don't have to round. Oh, look, a squirrel! *trick*
i'd have to disagree on that because the earliest number aren't of much interest if you want a precise value of the golden number.
We are talking about converging speed and we can see that in fact this series converge faster to the golden number than the fibonachi one.
thus you'd need less calculus to approach the rounded value to the n-th decimal to get it hence its usefulness.
edit: though a bit of mathematic rigor would be welcomed as his demonstrations reminds me of how i did maths in HS...
recouer, actually it's about elegance I think. As how much less precise calculus than (1+sqrt(5))/2 (which is exactly phi) do you want?
200 years ago the title would be an enigma
So the biggest take away from this is closing your eyes and rounding in prayer will give you any set of numbers you like to fit any argument! See I can maths two!
"I should give him directions to, the nearest... maths department"
I have not had such a strong or sustained laugh in quite a while sir. Bravo.
Fibonacci numbers are just Parker Lucas numbers
Thank you for upstaging the ubiquitous and apparently obligatory Rubik's Cube with Set and the best (you heard me) port of Asteroids.
Hey D, your kohai wants to get back in touch. Street sweeper baby...
You made a mistake! phi^n is not equal to F(n+1)+F(n-1), it's only approaching at +inf. So Fibonacci is still better.
But that is also true about Fibonacci: 1, 1, 2 . . . 3rd term divided by the second is 2/1 = 2, not phi. They approach the actual golden ratio in the limit as the number of terms approaches infinity, because only then will the ratio between two (extremely large) integers begin to approach an irrational number.
My favorite part of this video is how "Parker Square merchandise" pops up in the corner right after Matt admits he was wrong
Watched the whole video and I have one question...
Where's the beef?
Sebastian Elytron 😂😂
on the parker grill.
Someone just rounded it down
Sebastian Elytron That has to come with a sequence known as Narayana's Cows (OEIS A000930) with a recurrence Cn = Cn-1 + Cn-3. The ratio between successive terms is approx. 1.4656. We could call that the Beefy ratio designated by the Greek character Moo. Moo^2 - Moo = 1/Moo
Watch part 2.
I love matt so much LOL
The Lucas numbers do NOT satisfy L_n = round(phi^n) for all n, since L_1 = 1 does not equal round(phi^1) = 2.
You know, in the earlier Lucas Numbers video, the rounding seemed a bit weird to me too, but when you point out here that it's equivalent to pretending that the much-vaunted Golden Ratio of the Fibonacci numbers is actually the exact ratio and not just the limit that it tends to, that makes that earlier video feel much more intuitive.
Johnny joestar knows the golden ratio
Sir Lagsalot Is this a fibonacci reference?
What a slow dancer
Responding to an exact argument by hiding your rounding errors? What a Parker rebuttal! :P
We did it reddit!
That was some nice math judo, Matt. Taking your opponent's argument and revealing that it is actually an argument in your favor.
But [phi^n/sqrt(5)] gives us the n-th fibonacci number.
Also, I'm not either camp, recurrence is the lord of them all.
As another thing in defence of the Fibonacci sequence being tied to the Golden Ratio, if you look at the continued fraction of the Golden Ratio and look at its convergents, then they are F_{n+1}/F_n where F_n is the nth Fibonacci number. Which means that, for example, 21/13 is the ratio of two consecutive Fib. numbers and you can't find a rational number closer to the Golden Ratio with a denominator less than or equal to 13. So F_{n+1}/F_n is the closest rational number to \phi relative to F_n (relative to F_n meaning that there doesn't exist an p/q closer to \phi if b is less than or equal to F_n). So, in that sense the Fibonacci sequence is the fastest converging sequence to \phi.
Where is zeproxy???
Are you here???
Your points for the Lucas numbers was a real parker square of an argument
Matt stop
You’re making just a Parker square of yourself
Whoever subliminally put in that Parker Square at 6:03...well played 😏