Lucas Numbers and Root 5 - Numberphile

With Matt Parker. This is a continuation from: • Golden Ratio BURN (Int...
The original trilogy of videos where this all started: bit.ly/GoldenTrilogy
More Matt Parker videos on Numberphile: bit.ly/Matt_Videos
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[@numberphile2]

Part 1 is at: kzbin.info/www/bejne/mZvbY2aXgsp1obM Check out some Numberphile T-Shirts and other stuff: teespring.com/stores/numberphile

[@tjreynolds685]

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?

[@losthor1zon]

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.

[@PC_Simo]

@losthor1zon I figured that equation out. Very nice solutions, indeed 🙂.

[@wyattstevens8574]

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.

[@MisterAppleEsq]

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.

[@General12th]

Brain like a supercomputer!

[@MrAthoOome]

Not complicated really, it's just a matter of solving a linear difference equation and that's pretty basic.

[@MisterAppleEsq]

+@@MrAthoOome True.

[@giuzeppeedreimeimban1019]

Hes my favourite math person of the parker squarish kind 😁

[@Surfinite]

It doesn't work though. It's wrong.

[@julian_ossuna]

A trilogy made of four parts ...a Parker trilogy.

[@nanigopalsaha2408]

Tell that to Douglas Adams

[@princyrathore1709]

a 4 sided triangle :)

[@olanmills64]

😂

[@PokerPlayerJames]

That's the first one of these Parker [item] Jokes that I've laughed at 🤣

[@ronniemouser9752]

Nearly spit out my drink

[@unvergebeneid]

"The golden ratio is the marketable version of root five." Matt does not get enough credit for this quote!

[@shruggzdastr8-facedclown]

It'll feature prominently in his next book, Humble Phi 😏

[@hariskayani4703]

"The golden ratio is the marketable version of the root 5" should be on a t-shirt

[@_1derscore]

marketale plushie

[@blacxthornE]

Parker Square flash at 7:28

[@audiocancer]

True marketing! 👌

[@montano0222]

Then the Parker Square T-Shirts link on top

[@41-Haiku]

I caught that and I was so happy.

[@nicholasleclerc1583]

“I [didn’t] get that reference”

[@00bean00]

WOW!

[@loreleihillard5078]

Is that a calculator still in the box in the background? New calculator unboxing video confirmed

[@dalitas]

You being wrong isn't a theme, it's a meme!

[@anononomous]

Themes and memes are both closely related, they just happen to be expressed by a different sequence of letters.

[@silkwesir1444]

Also, actually the term "theme" fits much better (though "meme" isn't exactly wrong either).

[@pronounjow]

It's a running gag!

[@silkwesir1444]

Jo Reven yes, that term fits even better than the other two.

[@thechemuns74]

Theme VS Meme is the new Lucas vs Fibonacci.

[@Dankey_King]

3:34 Love how Matt's password is "PrkrSqr"

[@chaschtestark7973]

go watch the video "the parker square" from numberphile, and you will get the joke ;)

[@ObjectsInMotion]

@@chaschtestark7973 you're the one who isn't getting the joke.

[@MahraiZiller]

Should we then call the golden ratio “root Phive”?

[@GerSHAK]

:D

[@QuasarRiceMints]

*slow claps*

[@st.paulmn9159]

I’m here cuz I think one brake light is fine

[@PC_Simo]

@Mahrai Ziller Indeed we should 😁. I was actually thinking the exact same thing.

[@PC_Simo]

@St. Paul MN Don’t you mean: ”phine”?

[@quahntasy]

It's so amazing to see Matt come up with formula from those numbers without using any computer or something.

[@ffggddss]

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

[@gwahli9620]

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.

[@ffggddss]

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

[@GerSHAK]

Sweet. :)

[@jessstuart7495]

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,...

[@TimMaddux]

Back in the day it was originally called the quadratic fivemula, until the root 5 haters ruined it for everyone.

[@BastienHell]

"six! who knew?" Laughing so much it's painful

[@WAMTAT]

We can dream.

[@recklessroges]

*knew

[@BastienHell]

Thanks, missed the typo

[@Zarrykotter]

Parker being brilliant as always! - still chuckling

[@rosepinkskyblue]

I’ve watched this too many times and laughed every time

[@robinsparrow1618]

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

[@moneym0ney]

That is actually quite awesome but only holds true if you explicitly round down.

[@steffahn]

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 ^^

[@PC_Simo]

@@steffahn Exactly.

[@TheGreatPurpleFerret]

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.

[@MushroomManToad]

I always wanted to be first for a Matt Parker video. But here I am, second... Guess that makes this a Parker Comment

[@johnchancey3941]

5:06 - "That's my birthday!" 😂😂😂😂😂😂

[@aspiringcloudexpert5127]

Whoever came up with the name "Golden Trilogy" deserves a cookie.

[@julian_ossuna]

Interestingly, this trilogy is made of 4 parts. A Parker trilogy.

[@TheDWZemke]

Sure we would have PI and the cookie number...

[@Ramiprops]

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.

[@fcturner]

Very nice 👌🏽

[@vittoriosellingerstiatti4619]

Cameraman: "Oh, that's my birthday :D" Matt: "there you go :)" Cameraman: "No, that's not" What a savage.

[@NikhilSingh42]

"If i say 'this is interesting' enough times, it will be" :D

[@Nossairito]

Root 5 is the Iniesta to Golden Ratio's Messi.

[@ig2d]

Or the Buzz Aldrin to the golden ratio's Neil Armstrong. (Buzz Aldrin, the only person famous for not being famous)

[@ericvilas]

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!

[@chrisg3030]

Another beautiful thing: (Phi^1)+1 = Phi^(1+1) and Phi+(1^1) = (Phi+1)^1. The golden bracket shift.

[@ericvilas]

@@chrisg3030 ooh! Well, the first one, anyway, the second one seems like it would always be true?

[@chrisg3030]

Dammit you're right. Ok, so just the first one.

[@matthewbertrand4139]

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.

[@chrisg3030]

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)

[@pigeonfog]

I love the nod to the parker square when matt messes up the 5.

[@defuncttobedeleted3675]

Beautiful maths in the previous video! Very pleasing arguments Matt :D

[@aok76_]

The general form is certainly pleasing. You're an inspiration Matt!

[@bunshichi93f]

"The Golden Ratio is the marketable version of sqrt(5)" Don Draper approves!

[@chrisg3030]

bunschichi93f I guess it would be sold in a spray can (sqrt)

[@yaj126]

4:02 Wow that's some advanced math you just did there off the top of your head

[@LeoStaley]

He never bothered memorizing the fibonaci or lucas numbers, but he clearly memorized the exact decimal values of the ratio between them. A true legend.

[@josephbentham3901]

Sneaking in a Parker square. Loved it!

[@honkynel]

Really enjoyed.

[@seedschi]

"Root five is more the Ghostwriter for the Golden Ratio...." absolutely fantastic :-D Very nice video, I enjoy your number-juggling a lot!

[@juustgowithit]

I love this video so much

[@jaipod4561]

Did anyone else notice the "blink and you miss it" Parker Square at around 7:25?

[@fluffywhiteducky]

"Its like the ghost writer for the golden ratio" Hahahahahaha

[@JordanMetroidManiac]

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.

[@jannieschluter9670]

I love the Golden Ratio and all and everything about it!

[@alicealice180]

I love Matt Parker

[@maartendas1358]

Probably no news to anyone who studies this but I just found this out, and I didn’t find any comment yet describing this so I decided to just put this out there. When you write down the Lucas numbers and start adding the first and third number, the second and fourth, fifth and seventh, sixth and eighth and so on, the sequence you get consists of multiples of fives, following the same pattern as the original sequence (each new number is generated by adding the previous two together). Like this: 5, 5, 10, 15, 25, 40… Kinda neat when you consider how root 5 is so essential to the Lucas numbers and to similar sequences.

[@adnanchaudhary5905]

"If I say interesting enough times it will be" goes for everything in maths mate

[@SuperYoonHo]

awsome!!!

[@atrumluminarium]

I love how everytime a Parker Square flashes on the screen we get a PS T-shirt card lol

[@EllieSleightholm]

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! :)

[@SillieWous]

woohoo Matt Parker!!!!

[@johnnye87]

The fact that 1/phi + 1 = phi makes it seem like the series starting with those three values should be significant somehow...

[@sander_bouwhuis]

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)

[@n.l.4025]

Finally, I now get to understand this and get to see the “Parker #5”! For everything, there is a Parker Something! (Eccl. 3:1-4)

[@captapraelium1591]

Φst

[@MisterAppleEsq]

You're closer to πst, really.

[@WhattheHectogon]

πth* c'mon, duh

[@captapraelium1591]

πrd* XD

[@98danielray]

fist :v

[@konstantinbachem9800]

eth

[@frankharr9466]

I think it's neat that with this formula, you can find an A and B for the Fib and Lucas numbers.

[@KaroxNightshade]

They put in the Parker Square for one frame. Anger.

[@shruggzdastr8-facedclown]

Also, if you sequentially subtract the Fib #s from their corresponding Lucas #s and divide those differences by 2, you get another set of Fib #s (offset by one step).

[@PC_Simo]

Root(5) is the core or the heart of the Golden Ratio.

[@chaschtestark7973]

the Parker square-flash killed me :D

[@JonathonV]

Of course there's a game of SET on the shelf behind Matt! I tried to explain that game to my math prof who did his doctoral thesis on Latin Squares, thinking he'd be cash at it, but actually he found it pretty difficult to follow. It's one of the few games where kids, en masse, tend to do BETTER than adults because their pattern recognition skills are at the forefront of their minds.

[@trudyneo]

The Parker sequence!

[@rebmcr]

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!

[@sander_bouwhuis]

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.

[@andrewjetter7351]

The subliminal Parker Square when he messed up writing a 5 was perfect.

[@n.l.4025]

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.

[@parreiraleonardo4189]

One more interesting fact about Lucas numbers (Ln) and Fibonacci numbers (Fn): Fn•Ln=F2n

[@DaTux91]

There should be a Parker Square somewhere in London.

[@NereosRenbur]

The timing of the popup of the Parker Square T-Shirt is mean :D (also the square itself there)

[@slightlokii3191]

That Parker square sublim at 7:30 hahahaha

[@Yupppi]

Congrats Matt Parker on doing Arbitrary numbers formula on the go!

[@dubsed]

Favorite thing about root 5 is that it is the hypotenuse of the 1, 2 triangle, which itself forms the dihedral angle of a dodecahedron.

[@captapraelium1591]

This series was apparently edited by a nerdy version of Tyler Durden.

[@recklessroges]

I think it was his brother Parker Durden.

[@Nate-zb4bk]

I am Brady’s complete lack of surprise.

[@ImAllInNow]

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.

[@ShaMan54321]

Nice Parker Square flash.

[@shruggzdastr8-facedclown]

(@ 7:28 ): You didn't "fudge" your sqrt5, Matt -- you gave it a go!

[@dfs-comedy]

Now we know who is secretly pulling all the strings.

[@Lily-zy6uk]

Had to slow it down to see the Parker Square XD

[@xCorvus7x]

Well, in a context where we're more concerned with the average between 1 and any other number than with that other number (and so much so that this average could replace the variable input that defines it), the Golden Ratio is the essence of sqrt(5).

[@MattMcIrvin]

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.

[@legendgames128]

2:31 a few seconds before you revealed it, I guessed it was root five since it was kinda square root of some number like, and it was closer to 4.

[@JoshuaHillerup]

I will be really impressed when Matt can make pi and/or e pop out.

[@andrewolesen8773]

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?

[@meisam9592]

This is better than the original video.

[@scimatar5]

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.

[@ethanjensen661]

7:29 LOL! Then I saw Parker square t shirts

[@thomassynths]

Where can I buy the T-shirt Matt is wearing?

[@davidgillies620]

A generalised Lucas sequence is x_n = A x_(n-1) - B x_(n-2). This gives you a characteristic equation of x^2 - A x + B = 0. Let D be A^2 - 4B and positive. Then the roots of the equation are a = (A + sqrt(D))/2 and b = (A - sqrt(D))/2. You can form two sequences of integers, U_n = (a^n - b^n)/sqrt(D) and V_n = a^n + b^n. For A = 1 and B = -1, U is the Fibonacci numbers and V is the Lucas numbers. But all choices of A and B for which D is positive converge on a ratio for successive values of a, which for the Fibonacci and Lucas numbers is (1 + sqrt(5)/2, the Golden Ratio.

[@WildEngineering]

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.

[@WAMTAT]

This is interesting.

[@pronounjow]

Parker 5!

[@PhilBagels]

Let's all root for root-5!

[@WarpRulez]

Root 2 gets all the glory. Root 5 doesn't get even nearly as much love as it deserves.

[@usernamenotfound80]

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.

[@cubicardi8011]

How did you make those integer or rational number number range characters?

[@usernamenotfound80]

ℚ and ℤ are in Unicode. I just copy-pasted the symbols from Wikipedia: en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols

[@cubicardi8011]

@@usernamenotfound80 thx

[@matthewbertrand4139]

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.

[@cubicardi8011]

@@matthewbertrand4139 yes! That's what i always searched for. Thanks alot

[@robogiraffe]

I was thinking 8 and 3 too! That's so weird XD

[@jelmerl1458]

Never forget the parker square

[@ThePoxun]

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?

[@westtech001]

Just imagine how the Pythagorians would have loved discovering this relationship!

[@alexpotts6520]

A small caveat - for these "arbitrary numbers", the formula doesn't actually work for small n. I imagine what's going on here is that the round error gets smaller and smaller for larger n, but for small numbers it's significantly larger than 1 and therefore just rounding doesn't get the job done.

[@willgrannis2652]

When is Matt's next book coming out?

[@etnamecul]

me love this vid

[@WaltRBuck]

The magic of root 5. The question is, "How is it related, mathematically, to 42?"

[@theDevintage]

Edit: I'm wrong, thanks to Vojtěch Strnad for the correction. But for the sake of learning, this is what I wrote: There are infinite amounts of 42's in the digits of root 5, though I suppose that's not hard for an irrational number.

[@edwardpaddock2528]

Root 42 x Root 5 = 14.4913767. The digits of 14.4913767 sum to 42. Ab ovo usque ad mala

[@vojtechstrnad1]

@@theDevintage Is there some proof of this? Because you might have fallen for a known misconception. There is no number that we know is normal, although a lot of irrational constants are conjectured to be and a proof can be given that "almost all" numbers are normal.

[@theDevintage]

Thank you for pointing this out, you're right. You can have irrational numbers where certain numbers don't show up, my bad.

[@vojtechstrnad1]

It has been conjectured that every irrational algebraic number is normal, that includes root 5. So you might actually be right. It's just that I hear people assume this is proven too often, mostly in saying "the decimal expansion of pi contains every possible sequence of numbers".

[@Xnoob545]

√5 - the real Doug φ - Carl

[@MusicFanatical1]

These guys and their flight-of-fancy analogies....

[@PhilosopherStone]

Is there a possibility to acquire that paper?

[@adammasters2207]

Why is there a nautical chart of the South Atlantic in the background at the start of the video? A retrospective of the Falklands war?

[@jackscrivens9520]

3:36 hidden stand up maths reference :)

[@suwinkhamchaiwong8382]

Great

[@nomekop777]

*praises Lucas numbers* "I've never bothered memorizing them"