Lucas Numbers and Root 5 - Numberphile
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- เผยแพร่เมื่อ 28 ก.ย. 2024
- With Matt Parker. This is a continuation from: • Golden Ratio BURN (Int...
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what if you didn't use root 5 for this? what if you did root 7 instead? root 6? what if instead of the square root, you used the cubic root? would you still be able to come up with some "fibonacci-esque" addition?
I ran into the root 5 relationship with the golden ratio back in high school, when I tried to figure out a very simple quadratic equation: x^2 - x - 1 = 0.
For a geometrical representation, check out the ratios of the lines in a pentagram. The larger lines to the smaller are in golden ratio proportions.
@losthor1zon I figured that equation out. Very nice solutions, indeed 🙂.
5:20 Is the formula for the generalized sequence related to how many lots of the Fibonacci and Lucas numbers make it up? The Fibonacci and Lucas numbers are complementary Lucas sequences.
I'm really impressed that Matt managed to work out the formula for the arbitrary numbers completely by himself without using any other kind of tool.
Brain like a supercomputer!
Not complicated really, it's just a matter of solving a linear difference equation and that's pretty basic.
+@@MrAthoOome True.
Hes my favourite math person of the parker squarish kind 😁
It doesn't work though. It's wrong.
Should we then call the golden ratio “root Phive”?
:D
*slow claps*
I’m here cuz I think one brake light is fine
@Mahrai Ziller Indeed we should 😁. I was actually thinking the exact same thing.
@St. Paul MN Don’t you mean: ”phine”?
"six! who knew?"
Laughing so much it's painful
We can dream.
*knew
Thanks, missed the typo
Parker being brilliant as always!
- still chuckling
I’ve watched this too many times and laughed every time
I always wanted to be first for a Matt Parker video. But here I am, second... Guess that makes this a Parker Comment
Another interesting note is that a right triangle with an angle of π/5 yields sides with lengths defined by the golden ratio. The golden ratio can be obtained with 2cos(π/5). How might the power series expansion of cosine help determine why π and Φ are related in this way? And by a hypothetical syllogism, Φ must also be related to e. We could say e^(5icos^-1(Φ/2))+1=0, which relates e to Φ.
The fact that 1/phi + 1 = phi makes it seem like the series starting with those three values should be significant somehow...
As the video already stated... it doesn't matter with which (positive) numbers you begin. They all tend to the golden ratio, and they all result in splitting the components in two Fibonacci series. The johnye87 numbers :
1/φ+1
φ
φ + 1(1/φ + 1)
2φ + 1(1/φ + 1)
3φ + 2(1/φ + 1)
5φ + 3(1/φ + 1)
8φ + 5(1/φ + 1)
13φ + 8(1/φ + 1)
I love the Golden Ratio and all and everything about it!
This series was apparently edited by a nerdy version of Tyler Durden.
I think it was his brother Parker Durden.
I am Brady’s complete lack of surprise.
Now we know who is secretly pulling all the strings.
One more interesting fact about Lucas numbers (Ln) and Fibonacci numbers (Fn):
Fn•Ln=F2n
I must disagree with the statement that φ is the marketable version of √5. Instead, I'd say that out of the two, φ is the fundamental one, whereas √5 is merely some fallout that you happen to get additionally. The main reason for this is that if you look at the field ℚ(φ) (or ℚ(√5), they're identical), its ring of integers is ℤ[φ] and not ℤ[√5]. For the same reason, I think that ζ₆ = (1+√-3)/2 (a root of unity!) is more relevant than √-3 and more generally (1+√d)/2 is more relevant than √d for square-free integers d that are congruent to 1 mod 4.
How did you make those integer or rational number number range characters?
ℚ and ℤ are in Unicode. I just copy-pasted the symbols from Wikipedia:
en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols
@@usernamenotfound80 thx
I've got a math keyboard called dxMath which has those symbols on it as well as many others, if you don't want to have to do that every time.
@@matthewbertrand4139 yes! That's what i always searched for. Thanks alot
Φst
You're closer to πst, really.
πth*
c'mon, duh
πrd*
XD
fist :v
eth
The most amazing thing I saw in this video is the divisor in the generalised formula.
That thing is a PURE ten: not an artefact of the base-10 counting system, but the real true value 10 in its own right. You don't see that very often!
I agree. That threw me for a loop. Because 10 has the factors 2 and 5 it almost always gets factored out. Most numbers (60%) are divisible by either 2 or 5 or both.
The subliminal Parker Square when he messed up writing a 5 was perfect.
Root 5 is the drummer, the golden ratio is the front man. Everyone knows the front man of a band, but smart people know that he wouldn't be there without the drummer. In fact, most of the band wouldn't be there without the drummer.
Amazing. I'm a university maths youtube vlogger and I can't tell you how much numberphile has helped and inspired me over the years! :)
3:36 hidden stand up maths reference :)
The Fibonacci Sequence is the base sequence for the Lucas Numbers and the other sequences featured here. The Fibonacci Sequence is the number 1 sequence with a basis of root 5.
Nice Parker Square flash.
*praises Lucas numbers* "I've never bothered memorizing them"
These guys and their flight-of-fancy analogies....
I love this video so much
The timing of the popup of the Parker Square T-Shirt is mean :D (also the square itself there)
Congrats Matt Parker on doing Arbitrary numbers formula on the go!
Matt, I can't believe you didn't mention that the Grafting Constant is also based on root five: 3 - sqrt(5). Also, not sure if you know this, but in base 5, there's another grafting constant (The only two perfect grafting constants for square roots in any base) and it's value is: (3 - sqrt(5)) / 2.
There should be a Parker Square somewhere in London.
What’s the formula for this sequence:1443,2337,3780,6117...
I think you can get rid of the rounding in all of these formulae by bringing in the powers of the other root of the characteristic equation, which here is -1/phi.
7:29 LOL! Then I saw Parker square t shirts
7:30 Parker 5.
Never forget the parker square
I won't show this...
...
He shows it.
or there is an exact formula for the fibonacci numbers :
Fn = [ ((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n ] / sqrt(5)
this is link to the recurring sequence Fn+2 = Fn+1 + Fn which is linked to the quadratic equation x² = x+1, for which the two solution are the golden ratio phi=(1+sqrt(5))/2 and the other one psi=(1-sqrt(5))/2, and the formula rewrites itsef Fn = [ phi^n - psi^n ] / sqrt(5)
Okay so idk if this means anything but if you try to mirror the numbers IE) after the 0th digit, make the -1st -2nd etc. They both become mirrored across the 0th digit but their signs alternate eachother.
Last time you called braddy sequence
In a quick scan of the comments, I didn't see anything about this...
There is no need to round when calculating Fibonacci numbers:
Fn = (phi^n - (-1)^n * phi^-n)/5^0.5
Of course, on a computer, you will need to round due to FP precision issues (eg, F4 = 3.000000000000004 instead of 3),
What's do the numbers flashed at 7:27/7:28 mean?
29^2 1^2 47^2
41^2 37^2 1^2
23^2 41^2 29^2
I also downloaded the video to have a look at the flashing subliminal message. It's a Parker message.
I see people are starting to ping the shiny armor of Matt’s persona. Well the Internet revealed to us that there are millions of great guitarists in the world. Maybe this is sort of like that. But I very much enjoy Matt’s positive attitude. And since we are all human, what piques my interest here is how the discrete operator is accomplishing all this. We see, but I am not sure we understand.
You can diagonalize a matrix to find this formula. It works, but is somewhat clunky.
Why did you bring linear algebra into this?
That's actually the standard way of computing this type of recursive sequences. Fibonacci sequence is just a particular example of it. If you take the matrix [[0,1][1,1]] and apply it to the vector with two consecutive terms [Gn,Gn+1] of the generalized sequence, you'll get [Gn+1,Gn+2]. If you diagonalize this matrix, you'll get the golden ratio and its conjugate as diagonal values. So linear algebra is actually a pretty neat tool for this.
I wonder what physical significance that matrix has. (Sorry for my memey comment)
All the rounding is unnecessary if you use both values of phi. (phi^n - ihp^n)/sqrt(5) = fib
Had to slow it down to see the Parker Square XD
Pick A=1, B=3
-->
No square roots in the formula, which makes it look nice and clean
-->
Sequence = 1, 3, 4, 7, 11, 18, 29... = Lucas numbers starting from the second position
They are actually the Lucas numbers starting from L₁. So, in your example, A = L₁ and B = L₂. Matt's general formula was for arbitrary A and B. Here's a formula with no rounding for any A and B you like. It's rather less succinct than Matt's formula, but it works for low values and even negative values of n.
Gₙ = ( ( φⁿ⁻³ + (1 ‒ φ)ⁿ⁻³ + φⁿ⁻¹ + (1 ‒ φ)ⁿ⁻¹ )A + ( φⁿ⁻² + (1 ‒ φ)ⁿ⁻² + φⁿ + (1 ‒ φ)ⁿ )B ) / 5.
Subliminal parker square
Of course I recognize the square root of 5. Come on, give me some credit!
Hey I was born on 97/65 too!
I love the Parker square at 8:28! Numberphile easter eggs are the best.
I've been wondering why does Fn=(φ^n-(-φ)^-n)/√(5) exactly;
Why does subtracting (-φ)^-n/√(5) from the identity do the approximation for us?
It has something to do with the fact that φ-1/φ=1 of course
Having read the first few chapters of things to make and do in the fourth dimension, I must ask have you generalized this to all bases?
PARKER SQUARE
What about root 3? Or root 7? How one could "market" them?
At about 1:57, you realize he's doing that in his head...
I don't know much about algebra but I take the credit for golden ratio. In fact, I'm gonna trademark it and sell T-shirts and coffee mugs.
so what happens if you switch out those sqrt(5) with something else.. say the square roots of other primes such as sqrt(3) or sqrt(7)? anything interesting?
7:28 - Parker Square
So, root five is like, the Parker number?
7:28 👌👌😂😂
That is definitely a Parker equation for the "arbitrary" numbers.
In that it does not work. How Matt thinks an equation like that could possibly give you 8 then 3 then 11 is beyond me.
Golden ratio p= sqrt(5), where "p=" is Parker equality a/k/a "pequals."
*STERN-BROCOT* >>> _everything._
7:27 Parker's five
7:28 Parker square :)
F(n) = (phi^n - (1-phi)^n)/sqrt(5). Exact, no rounding
what's the generalized form of this for other sequences?
So if we're talking about sequences z(k) where the rule is add two terms to get the next one (z(n) = z(n-1) + z(n-2) for n > 1)
Let p = phi and q = -1/phi
(there's plenty of useful simplifying identities with these, such as p + q = 1, p - q = √5, p*q = -1, p + 1 = p*p, q + 1 = q*q, etc)
If the first two terms are z(0) = a and z(1) = b, then the general term is z(n) = Ap^n + Bq^n
with A = (b - qa) / √5 and B = a - A
So using that formula,
For the Fibonacci numbers starting 0, 1, ... we get (A, B) = (1/√5 , -1/√5)
For the Lucas numbers starting 2, 1, ... we get (A, B) = (1, 1)
And for the "arbitrary" numbers starting 8, 3, ... we get (A, B) = (4 - 1/√5, 4 + 1/√5)
(writing it this way shows that there's probably a prettier way to write my formula above, since for integer case it should always come out looking like this)
BTW, I think it's useful to look at these sequences as a vector space. If we define fib(-1) = 1 (this is consistent with the recurrence) then fib(n-1) [1, 0, ...] and fib(n) [0, 1, ...] are a convenient basis. Looking at it this way we can see that a sequence starting a, b, ... should be equal to a*fib(n-1) + b*fib(n)
Right, so the more symmetric form of the formula is, for sequence starting a, b, ...
we can write A = (a/2) + k and B = (a/2) - k, where k = (b - a/2) / √5
This version is nicer when a, b are integers (little to no simplifying)
7:28 ?
Who saw the parker square at 7:28
√(5)=Phi+Phi^-1
I can here for philosophy and all I got was this lousy t-shirt (Parker square)
May i know... What is the benefits on this?
7:28 What is going on?
I love Matt Parker
What do you call the Sum Numbers resulted from Fibonacci+Lucas Numbers?
F + L = S
1 + 1 = 2
1 + 3 = 4
2 + 4 = 6
3 + 7 = 10
5 + 11 = 16
8 + 18 = 26
13 + 29 = 42
21 + 47 = 68
...
A double Fibonacci I guess, especially as you could have started with 0 + 2 = 2
i am the 177the viewer hehehe
i see hehe classic parker square argument tho "the golden ratio is √5 _with a little fudging_
the pfp for this channel should be √2 tho :3
~parker fan
Wait, are you now an actual fan, or not?
only actual fans will get it :3
I guess, then I am a Parker Fan.
not all actual fans are parker fans and not all parker fans are actual fans
heck not all parkers fans are parker fans
hence we are Parker Fans /o/
@@fuseteam
One shot for every time someone in this comment thread says Parker Fan.
To be sure, one for every other word as well.
Would be nice if you'd, for once, talk about the Cordonnier numbers (aka Padovan numbers), Perrin numbers, Van der Laan numbers, and the number that ties all three together, the only other morphic number besides the golden ratio: The plastic number - albeit I prefer the french name, le nombre radiant, "the radiant number". The argument about Fibonacci vs. Lucas transfers to Cordonnier vs. Perrin (but not vs. Van der Laan, which, as a third sequence, only becomes relevant when you deal with the plastic number.). See here, for example, for details: dx.doi.org/10.1080/00207390600712554
Also, interestingly enough, the "arbitrary" sequence you wrote down returns a result when looked up in OEIS:
oeis.org/A152459
It's not exactly that sequence, but close enough - and related to the Fibonacci numbers in a **different** way.
2 square plus one square. That is square root 5. Relationship between two and one. Bends over one and two. The smallest curves of non equality of nature. So golden ratio is a derivative of Lucas. First force vector.
so John Davis, Brad Howell, Charles Shaw are root5 and Milly Vanili is golden ratio.ok.
A trilogy made of four parts ...a Parker trilogy.
Tell that to Douglas Adams
a 4 sided triangle :)
😂
That's the first one of these Parker [item] Jokes that I've laughed at 🤣
Nearly spit out my drink
"The golden ratio is the marketable version of root five." Matt does not get enough credit for this quote!
It'll feature prominently in his next book, Humble Phi
😏
Is that a calculator still in the box in the background?
New calculator unboxing video confirmed
"The golden ratio is the marketable version of the root 5" should be on a t-shirt
marketale plushie
You being wrong isn't a theme,
it's a meme!
Themes and memes are both closely related, they just happen to be expressed by a different sequence of letters.
Also, actually the term "theme" fits much better (though "meme" isn't exactly wrong either).
It's a running gag!
Jo Reven
yes, that term fits even better than the other two.
Theme VS Meme is the new Lucas vs Fibonacci.
Parker Square flash at 7:28
True marketing! 👌
Then the Parker Square T-Shirts link on top
I caught that and I was so happy.
“I [didn’t] get that reference”
WOW!
The *real* reason that √5 keeps popping up in "Fibonacci-esque" sequences is the iteration rule coefficients,
A₊ = 1·A₀ + 1·A₋₋ ; i.e.,
1·A₊ - 1·A₀ - 1·A₋₋ = 0
and the corresponding quadratic that comes from that:
x² - x - 1 = 0,
whose discriminant is
b² - 4ac = 5,
the radical of which appears in the quadratic solution.
If you generate a sequence with different iteration coefficients, you will get a different limiting ratio, which is a zero of a different quadratic, with a different discriminant.
Which might in itself, make an interesting further addendum to these two videos. How to make a "designer" pyrite* ratio . . .
* pyrite = "Fool's Gold"
Fred
And because it's the solutions to a quadratic equation, there are actually TWO golden ratios (sqrt(5)-1)/2 is too. Which caused some confusion for some people but it just means that as a ratio 2:1 is the same as 1:2 or 1:1/2 ... simply that the bigger value is twice as big as the smaller one.
Well, actually, the other solution is negative.
x² - x - 1 = 0
x = φ = ½(1 + √5) = 1.61803..., and x = ½(1 - √5) = -1/φ = 1-φ = -0.61803...
In the Fibonacci sequence, the ratio of consecutive terms, Fᵢ₊₁/Fᵢ → φ as i → ∞,
and Fᵢ₊₁/Fᵢ → -1/φ as i → -∞
Fᵢ = ... , -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Oh, and the bigger solution is -φ² = -(φ+1) = -2.61803... times the smaller one, not twice.
Fred
Sweet. :)
How about A₊ = 3·A₀ - 1·A₋ That gives you a discriminant of 5 also.
I vote for calling (3 + sqrt(5))/2 = 2.618 the "Pyrite ratio".
Interestingly if you calculate this sequence starting with 1,1 You get every other Fibonacci number.
1,1,2,5,13,34,...
Back in the day it was originally called the quadratic fivemula, until the root 5 haters ruined it for everyone.
It's so amazing to see Matt come up with formula from those numbers without using any computer or something.
i actually memorized the first 9 digits of root 5, because of my interest in phi
interestingly the first 9 digits of root 5 + 1 contains three palindromes in a row
3.23 606 797
which is one of the things that made it fairly easy to memorize
That is actually quite awesome but only holds true if you explicitly round down.
since they're saying "the first 9 digits are," and not "the number, rounded to 9 digits, is," I don't think there's anything one needs to be more "explicit" about missing, except for (maybe) use base 10 ^^
@@steffahn Exactly.
3:34 Love how Matt's password is "PrkrSqr"
go watch the video "the parker square" from numberphile, and you will get the joke ;)
@@chaschtestark7973 you're the one who isn't getting the joke.
The expression he gave further simplifies if you substitute √5=2φ-1 (golden ratio definition with the √5 isolated) in all the numbers which have a √5 factor (including the 5s and one from the 10 in the denominator, since 10=2*√5^2) and becomes G_n= [φ^n*((2-φ)A+(φ-1)B)/(√5)], which is much much more visually appealing than the expression Matt gave.
Very nice 👌🏽
Whoever came up with the name "Golden Trilogy" deserves a cookie.
Interestingly, this trilogy is made of 4 parts. A Parker trilogy.
Sure we would have PI and the cookie number...
Root 5 is the Iniesta to Golden Ratio's Messi.
Or the Buzz Aldrin to the golden ratio's Neil Armstrong. (Buzz Aldrin, the only person famous for not being famous)
The big reason why many mathematicians like Fibonacci numbers and phi is because of its continued fraction: 1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(...))))))). I'd say that reason alone makes it a big deal.
Another thing: when you talk about the golden ratio, you should talk about 1-phi as well! I'd definitely consider it important enough that you should give it another name. Let's call it chi. Chi is also phi's conjugate, its negative reciprocal, AND, what do you get when you subtract chi from phi? The square root of 5!
So, phi and chi combined encapsulate all the beauty that people give to the golden ratio, all the beauty of the Lucas AND Fibonacci numbers, AS WELL AS all the beauty of the square root of 5.
I love those 2, phi and its little brother, being all mathy and stuff.
EDIT: Another reason why you should've talked about chi: you don't need to make approximations and your precious Lucas numbers always look a lot neater!
L(n) = phi^n + chi^n, exactly, always.
F(n) = (phi^n - chi^n) / (phi - chi), also exactly.
Ooh, you should make a video about ways to make non-radical numbers by using conjugates!
Another beautiful thing: (Phi^1)+1 = Phi^(1+1) and Phi+(1^1) = (Phi+1)^1. The golden bracket shift.
@@chrisg3030 ooh! Well, the first one, anyway, the second one seems like it would always be true?
Dammit you're right. Ok, so just the first one.
Well, since φ - χ is equal to √5, I would argue it should be expressed in that form in the denominator of your rule for F(n), but that's otherwise amazing, man. I do get that you were just emphasizing their prevalence in these rules, but for practical purposes I'd use the radical.
Eric Vilas Popular name: Geometric, Fibonacci, Narayana First few terms: 1 2 4 8 16 32, 1 1 2 3 5 8 13, 1 1 1 2 3 4 6 9 Recurrence relation: An = An-1 + An-1, An = An-1 + An-2, An = An-1 + An-3 Ratio value: 2, !.618. . ., 1.4656. . . Designation: 2, Phi, Mu Bracket shift equation: (2^0)+1=2^(0+1), (Phi^1)+1=Phi^(1+1), (Mu^2)+1=Mu^(2+1)
I love that Matt pointed out where the rounding was hidden in the last video. I never would have caught that and honestly it helps explain this a whole let better just with that tidbit.
"If i say 'this is interesting' enough times, it will be" :D
Cameraman: "Oh, that's my birthday :D"
Matt: "there you go :)"
Cameraman: "No, that's not"
What a savage.
5:06 - "That's my birthday!" 😂😂😂😂😂😂
How about instead of writing ((3√5 - 5)A + (5 - √5)B)/10 instead write (3A - B)/2√5 + (B - A)/2 ? I guess the former gets to the amount of A and B quicker, but the latter looks into the relationship between the two.
so (3A-B)/2√5=(A-B)/2? which would mean 3A-B=(A-B)√5..........oh my
@@fuseteam wtf are you talking about... That's not at all what OP implied
i simply did some maths!
@@fuseteam ok, I'm sorry for being rude... I thought you meant that OP had made a rearranging mistake. You can prove that ur last line is correct by multiplying both sides by zero :p
lol and thanks for being sincere and verifying :3 -also multiplying both sides by zero even proves 1=2 :P-
what i did was assume that the golden ratio-esque sequence number that we are looking for is 0 aka
(3A-B)/2√5+(B-A)/2=0 in which case we get 3A-B=(A-B)√5 which if we work it out further brings us 3A-B=A√5-B√5 so that (3-√5)A=(1-√5)B which is interesting :3
"The Golden Ratio is the marketable version of sqrt(5)"
Don Draper approves!
bunschichi93f I guess it would be sold in a spray can (sqrt)
I love the nod to the parker square when matt messes up the 5.
Did anyone else notice the "blink and you miss it" Parker Square at around 7:25?
Beautiful maths in the previous video! Very pleasing arguments Matt :D
4:02 Wow that's some advanced math you just did there off the top of your head
"Its like the ghost writer for the golden ratio" Hahahahahaha