The Silver Ratio - Numberphile

The silver ratio (and other metals) with Tony Padilla.
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[@SpeakShibboleth]

why use scissors? just use a numberfile.

[@graduator14]

This joke is the pinnacle of this channel! We can all go home now. :^p

[@poisonpotato1]

Under appreciated pun

[@Sunshine11229]

oof.

[@asad210]

Nice.

[@tylerhecht3360]

Ba Dum Tss (seriously, though, that was a legendary pun)

[@cuitaro]

sqrt(2): _happily exists irrationally_ Tony: now this *ratio*

[@TNTPablo]

The animator does an incredible job!

[@thesuomi8550]

Thanks :)

[@Stilllife1999]

Thanks. (it was me)

[@eyflfla]

Yeah, numberphile's animations have gotten better over time. I also like how they're low key and in that same brown-paper hand-drawn style.

[@nowonmetube]

@@thesuomi8550 no thanks, me it was!

[@nowonmetube]

@@Stilllife1999 no, me x)

[@kevinmackie4045]

Can't wait for the Bronze ratio and the Honorable Mention ratios :)

[@dielaughing73]

Participation ratios

[@jwcfive7999]

Can’t forget the steel ratio

[@geraldsnodd]

🤣

[@myboatforacar]

I'm holding out for the tin ratio

[@RWBHere]

Then there's the CdB ratio. No; not Cadmium Boride: It's the 'Could do Better' ratio.

[@distraughtification]

That sounds like a way to count seconds. One-bonacci, two-bonacci, three-bonacci...

[@maishamohiuddin297]

tb to the americans cant count video

[@otakuribo]

_one-bonacci_ _two-bonacci_ _red-bonacci_ _blue-bonacci_

[@MushookieMan]

Count von Count says, One-bonacci, two-bonacci, three-bonacci, AH AH AH AH!

[@klaxoncow]

One-bonacci, two-bonacci, three-bonacci, four. Five-bonacci, six-bonacci, seven-bonacci, more.

[@assassin01620]

One-bonacci, two-bonacci, three-bonacci, four. Four bonaccis make a metallic ratio and so do many more!

[@jeremyheminger6882]

"Metallic Ratio" is the name of my new Tool tribute band.

[@MisterAppleEsq]

Nice.

[@fossilfighters101]

Woah Mister Apple what're you doing in this comment section?

[@Pfhorrest]

Are you going to produce n-bonacci variants of Lateralus?

[@clyde8759]

Jeremy Heminger lol

[@MisterAppleEsq]

+fossilfighters101 I mean, right now I'm replying to your comment.

[@Artifexian]

Amazing! Had no idea these existed.

[@cuzeverynameistaken1283]

Had no idea you watched numberphile. Ive been following you since 4000 subs

[@harry_page]

Hey Edgar!

[@Artifexian]

Yup! Numberphile is one of my favourite channels.

[@natheniel]

It's one of everybody's favourite channels!

[@unvergebeneid]

PBS Infinite Series had a video on these. If you like Numberphile, you'll probably also like them!

[@matin563]

I also like the fact that the golden ratio is pronounced as *fi* (phi) and it can be found in the *fi* bonacci sequence

[@damianzieba5133]

That's why it is called fi

[@EtienneBotek]

@@damianzieba5133 phi is for Φειδίας

[@PC_Simo]

So do I 😎. Φbonacci.

[@alexandermcclure6185]

@@PC_Simo Shouldn't it be φbonacci instead of φibonacci? φ sounds like fi, not not f.

[@PC_Simo]

@@alexandermcclure6185 True. I think my train of thought changed, mid-word. 🤔😅 *EDIT:* I made the correction ✅😌👍🏻.

[@brnsndwch]

6:40 P=NP solved

[@efulmer8675]

Hah. That made me laugh.

[@akshat9282]

*REPETITION REPETITION REPETITION*

[@3ckitani]

REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION

[@cavorkehl6777]

REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION

[@prasanttwo281]

REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION REPETITION

[@Ready4Music]

Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition Repetition

[@CarelessMiss]

Akshat K Agarwal *REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION REPITITION*

[@flymypg]

I especially love Numberphile videos that provide generalizations, revealing the wider mathematical landscape extending from and encompassing a better known starting point.

[@jamezer7revel471]

The ratios are winning some medals here

[@Advection357]

Just the gold counts.. the rest are participation trophies

[@thelastspartanS117]

Hopefully no one MEDDLES in the award ceremony

[@thelastspartanS117]

Im sure the ratios have the Mettle to withstand such buffoonery

[@anticorncob6]

It’s Gold = 1 Silver = 2 Bronze = 3 Just like medals in the Olympics.

[@debayanbanerjee]

Jamezer7 Revel I think you meant 'metals'. Not medals. ;-)

[@AdeptAscent]

This video and the golden ratio (why so irrational) video were the most fascinating two videos I have ever seen on this channel. I hope you guys do more videos on these metallic ratios and how weird they are

[@markiyanhapyak349]

They aren't weird; They are constant.

[@WolfWalrus]

Of course, the Golden Ratio has the special property of allowing [Infinite Spin] according to the ancient Zeppeli family technique

[@zanly5039]

and the silver ratio allows for the almost-infinite spin

[@rubenbohorquez5673]

@@zanly5039 ah yes, the TREE(3) spin, not infinite, but stupidly big!

[@roskataka2574]

I see what you did there, fellow JoJo fan :

[@ramiroseltzer5278]

the Golden Ratio also allows cripples on a horse and a wheelchair to walk... its amazing what math can do

[@yazanabdalluh6251]

@@zanly5039 no its for polnareffs silver chariot to spin

[@matthijshebly]

The continued fractions are cool too and worth a mention: Golden ratio: 1 + 1 / (1 + 1 / (1 + 1 / (…))) Silver ration: 2 + 1 / (2 + 1 / (2 + 1 / (…))) Etc. Furthermore, you could expand into real numbers, with e.g. 3/2 giving an alloy of Gold and Silver, i.e. Electrum: 0, 1, 3/2, 13/4, 51/8, 205/16, 819/32, … which quickly converges to a ratio of 2. Let's call 2 the Electric Ratio. The numerators of the fractions follow an interesting pattern: 3 * 4 + 1 = 13 13 * 4 - 1 = 51 51 * 4 + 1 = 205 205 * 4 - 1 = 819 Etc.

[@TheInselaffen]

Britain; home of the Aluminium Falcon.

[@halyoalex8942]

Sounds like an odd crossover of Iron Man and Falcon from Marvel...

[@SteamPunkLV]

omg I hate when my nails looks like goddamn polygons xD

[@ziquaftynny9285]

Yeah, I didn't know people trimmed their nails like that. I usually cut them by the side then tear the rest off.

[@munro22]

Ziquafty Nny that’s not human

[@ziquaftynny9285]

No u

[@EchoHeo]

Ziquafty Nny yes me

[@UnderscoreZeroLP]

I bite my nails. So much easier

[@leoangere5310]

"Copper, nickel... aluminium?" That one cracked me up. Awesome content as always. I'll have to use these metallic ratios in my photo cropping (I've used the golden rectangle but then defaulted to boring ratios like 1:2, 1:3, etc.)

[@pakfu]

I’ve been on YT for like 12 yrs and this ranks in one of my favourite videos ever. Thank you so much.

[@SNNTV3000]

surely then, the 49° one should become the peregrine ratio?

[@fossilfighters101]

Ooh yeah!

[@TheOzumat]

It needs to be named after a metal. Aluminium is actually really nice for this purpose, as it's associated with aviation, which in turn is associated with birds.

[@theRealPlaidRabbit]

Or "Pippin" for short.

[@benjaminmiller3620]

Compromise and call it the Aluminium Falcon ratio?

[@rewrose2838]

@@benjaminmiller3620 Peregrin-inium

[@angst_]

So, I love the format of your videos! Someone who's passionate about something explaining it to the viewer/Brady as if just having a conversation. Brady seems to talk juuust enough and asks the perfect questions to make the conversation flow. Plus these recent animations are top shelf art!

[@OlbaidFractalium]

Wow! I did not know The Great Wave of Hokusai is geometric designs.

[@WildAnimalChannel]

yeah, if you fudge the results enough.

[@donaldasayers]

Looks more like a dragon curve to me.

[@pleindespoir]

Hokusai >>> Hokuspokus

[@BainesMkII]

Yeah, there was a *lot* of fudging required to make that painting fit the desired spirals. The best match was the middle spiral, and even there they had to cheat by jumping from the inside to the outside of the wave to get an overlap that ran for more than two and a half "squares". To fit the big spiral, they had to use two completely separate waves, half the length of the spiral matched nothing, and half of one of the waves didn't match. The small spiral didn't match at all; you could have claimed numerous random shapes matched as well as that small spiral.

[@genghiskhan6688]

Isn't it Kanagawa?

[@nakamakai5553]

As a marine biologist, I love these. Forms like this pop up all over the undersea world, especially among invertebrates. Well done!

[@WhiteChocolate74]

Ocean studies are underrated 💙

[@jerombastiaansen9495]

7:20 That's the solution to x² - Nx - 1

[@ultra6334]

Wow, makes so much sense, as phi's value is x^2 - x - 1, you just multiply that degree 1 x with some number to get these ratios

[@DeathlyTired]

All of these ratios are very important, integral even, to design in modern artisitic origami; especially the most usual kind that develops from a single, uncut square, to the finished model. You could perhaps talk to luminaries of the field such as Dr. Robert Lang for the intersection betweeen mathematics, origami, and it's real world applications.

[@markiyanhapyak349]

Wow!

[@unknown360ful]

Prof. TONY! The ratio videos are awesome!

[@manueldelrio7147]

Love Tony's videos!

[@whjk83921]

Fantastic episode! One of my absolute favorites!

[@torin1006]

I love things that use root(2) anywhere in them.

[@void9720]

1.41421356237309504880168872420969807856967...

[@wanderingrandomer]

Ahh, this makes more sense People would always overlay the golden ratio spiral over everything, even when it didn't fit, and it never made any visual sense to me. Now I know why... long story short, idiot conspiracy theories who know nothing about maths have been misleading me to the nature of logarithmic spirals.

[@MarvinFalz]

The golden ratio also appears in photography, which I wouldn't call idiotic nor conspiratory, but maybe in need of aditional information. But I would call New Age idiotic, since some New Agers use the fibonacci sequence as well as elements of quantum physics as proofs for their New Age teachings.

[@mwu365]

your short story was as long as your long story

[@Homs86]

same here. woodworking school is kinda obsessed with the golden ratio bcs"so pleasing" blabla finally there is light :)

[@bruhmoment1835]

Well, it IS a JoJo reference

[@popogast]

I'm always enlightened by the enthusiasm You mathematicians on this channel have. ITs a delight. Thank You.

[@fabianr253]

16:10 How I hypnose myself to stay consistent at learning

[@thatsleepytitan769]

A ratio for every element

[@Henrix1998]

Golden ratio should be hydrogen ratio then

[@Smittel]

S U L P H U R S P I R A L

[@briandiehl9257]

Yes

[@kuro13wolf]

Except bronze, which is an alloy. It's quite upsetting when you think about it.

[@Smittel]

Rhyme Bito copper turns green tho. You don't want a green medal do ya?

[@benjaminolanderrasmussen3049]

What would the ratio for the phi-bonacci sequence be called?

[@ericluque6573]

i was thinking the exact same thing

[@benjaminolanderrasmussen3049]

Eric Luque. When you remember the numberphile videos that you have recently watched :)

[@heyandy889]

the very golden ratio

[@primarysecondaryxd]

The golden-golden ratio. Plug it in to (N+sqrt(N^2+4))/2 - > The golden golden golden ratio, plug it in to (N+sqrt(N^2+4))/2 -> The golden golden golden golden ratio. Etc.

[@BubbaJ18]

Or π-bonacci?

[@yogitshankar6348]

Great to see Tony Padilla back!! Love the ratio videos

[@captainroll]

I love his enthusiasm for everything!

[@rider2fois]

An interesting thing about logarithmic spirals is that you can use them to define the analytic extansion of the zeta function.

[@cbbuntz]

I noticed some similar properties to the silver ratio to the golden ratio a while back. 1 / (2^0.5 + 1) = 2^0 .5 - 1 1 / ( 2^0.5 - 1) = 2^0.5 + 1 and a few others.

[@3c3k]

This is not related to the ratios

[@cbbuntz]

@@3c3k Actually it is. It's related to pell number generation

[@3c3k]

@@cbbuntz Have you not learned surds in school?

[@acerovalderas]

Excellent extension of the Golden Ratio. I love it!

[@iAmTheSquidThing]

Pete's animations often elevate Numberphile videos into something beautiful as well as informative.

[@AnthonyYandow]

I really enjoyed watching Brady with the camera in the window reflection! Neat little "behind the scenes included"

[@MasterHigure]

They're not really logarithmic spirals, though, are they? A true logarithmic spiral isn't piecewice circles.

[@Tumbolisu]

The formula is correct, but the whole "circles inside squares" thing is just an approximation.

[@stevethecatcouch6532]

The spiral he drew was the golden rectangle spiral, not the golden spiral. Another spiral that approximates both of them is the Fibonacci spiral, in which successive Fibonacci rectangles are used in place of the golden rectangle.

[@chrisg3030]

Dr Gerbils But isn't each of those successive Fibonacci rectangles, created each time a square is added, itself a golden rectangle, that is one whose aspect ratio is golden?

[@stevethecatcouch6532]

Chris G, No, the aspect ratio of a Fibonacci rectangle is only approximately the golden ratio. For example, 13/8 = 1.625, not 1.618 ...

[@chrisg3030]

Dr Gerbils I think I get it. "A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship (a+b)/a = a/b = Phi" (Wikipedia). Rectangles with Fibonacci number sides only approximate to this relationship. But if true golden rectangles were successively formed in this structure instead, what kind of spiral would result?

[@prathameshsundaram7509]

I love your channel! It's absolutely amazing!

I love using golden section in music. I learned so much about this studying Bela Bartok scores back in the 70s and 80s.

[@ThePrimevalVoid]

Man, the animations are getting trippier by the video.

[@kinshukdua]

OMG I just learned about the Silver ratio in Persona 5 and Numberphile uploaded a video about it, am I on something lol?

[@zyrota4295]

I love watching this channel because it makes you feel as if you stopped by the maths nerd's office and they just started to explain to you this cool math thing.

[@Banana_Split_Cream_Buns]

So the Pell sequence features 13^2=169, which is interesting as the Fibonacci sequence features 12^2=144.

[@mioszchrzempiec4429]

9:10 I never thought that watching a numberphile episode would be useful in persona 5

[@garrettkrawczyk9414]

What about a super metallic ratio where the ratio is between the golden ratio & silver ratio, silver ratio & bronze ratio, etc.

[@chrisg3030]

Using the formula (n + sqrt(4 + n^2))/2, so that when n=1 we get the golden ratio, and when n=2 we get the silver, then when n=1.5 we get the ratio exactly 2. Now if we construct a regular figure with the number of sides equal to the number under the radical, then it would be interesting to look at a figure with 6.25 sides to compare diagonal lengths on and see if any of them are in an exact 2:1 ratio, just as you get a silver ratio for a similar operation in an octagon. How would you interpret that? I tried a hexagon with a side produced by a quarter beyond the join with the next.

[@RaunienTheFirst]

@@chrisg3030 when I did the calculation for what I'm calling the half-bonacci, i.e. where N=0.5, I get the ratio to be (1+sqrt17)/4 Not sure where you got 2 from

[@chrisg3030]

RaunienThe First I got the denominator 2 from the formula at 7:21. I plugged in 1.5 in place of N since this value is half way between 1 (plugging in which gives you the Golden ratio) and 2 (plugging in which gives the silver ratio), and seemed to be what Garret Krawczyk was asking for, rather than the half-bonacci of 0.5. So with mine we get (1.5 + sqrt(4 + 1.5^2))/2 which gives a sequence ratio constant of exactly 2. Moreover for the Golden ratio the number under the radical in the formula is 5, for the silver it's 8, but for this intermediate case it's 6.25, so I was (not quite seriously) imagining a figure with 6.25 sides. Your figure would have 17 sides which sounds interesting..

[@geoffroi-le-Hook]

so an 18-Karat ratio ...

[@Ruxinator]

Great video! I particularly enjoyed this one

[@zanti4132]

Interestingly, the odd entries in the sequence for the Silver Ratio are the large numbers (i.e. the diagonals of the right triangles) for all the Pythagorean Triples where the two smaller numbers (the legs of the triangles) differ by one: 0^2 + 1^2 = 1^2 3^2 + 4^2 = 5^2 20^2 + 21^2 = 29^2 119^2 + 120^2 = 169^2 696^2 + 697^2 = 985^2 etc. Furthermore, you can generate all these Pythagorean Triples by selection the two consecutive entries in the Silver Ratio and applying that m^2 - n^2 / 2mn / m^2 + n^2 formula to generate Pythagorean Triples: m = 2, n = 1: Generates 3-4-5 m = 5, n = 2: Generates 20-21-29 m = 12, n = 5: Generates 119-120-169 m = 29, n = 12: Generates 696-697-985 etc.

[@dkamm65]

Could you not use this Pell Sequence to find very large primes? Since the numbers in the sequence grow exponentially faster than the position, couldn't you calculate the number in the (very large prime)th position to find a gargantuan prime?

[@user-ct1ns6zw4z]

The 7th pell number is 169, which is 13^2. All pell primes have prime indexes, but not all prime indexes correspond to pell primes. You might call them "pell pseudoprimes".

[@nowonmetube]

I thought the same thing, and another person besides you in the comments section as well. If there isn't a sequence that could find prime numbers. But if there is, we surely still haven't found it yet.

[@properbeatz]

Im 27 years old and I just found out what the metal part of a ruler was for... Thanks Numberphile!

[@nowonmetube]

How did you not think about that yourself

[@Difulsif]

The animations are getting better in each new video :D

[@amydebuitleir]

The animation on this video is really satisfying!

[@philosofickle]

Damn the ending was hilarious 😂😂

[@someweeb3650]

"We can easily work out how much you've cut off" You didn't have to explain anything for me to know the answer- too much.

[@LMacNeill]

It's just so fascinating how mathematics show up literally *everywhere* you look! Of course I've seen these spirals everywhere, but I've just never though about how you could describe them using mathematics. Fascinating!

[@angry4rtichoke646]

This person is awesome! I need to find more videos of them!

[@saidatulhusna1533]

i haven't watched the video yet but i assume this is about some sort of a parker ratio

[@jogiff]

I legit thought it would be about electrum or the gold standard

[@CaseyShontz]

Saidatul Husna not really lol but that’s what I thought

[@steph_dreams]

I like the Parker ratio but I prefer Parker squares

[@quantumhorizon]

Interesting video! I'm curious though, have imaginary analogues to the metallic ratios been explored?

[@G8tr1522]

ooo, interesting. Could you iterate a sequence of imaginary numbers?

[@user-pq7dy9op6y]

Beautiful fractal geometry!!!

[@stevefrandsen]

Very informational Tony. Thank you.

[@caiheang]

Does the Metallic Ratio Spirals have arc-length limits, or are they "infinitely long"? :O Like 1/2 + 1/4 + 1/8 + ... tends to 1 after infinite iterations. Does someone know the answer?

[@mxpxorsist]

The arc length of a quarter circle is pi/4*r where r is the radius. Therefore the arclength of a spiral with ratio 1/delta is (starting with r=1) pi/4*(1+1/delta+1/delta^2+...)=pi/4*delta/(delta-1)

[@badrunnaimal-faraby309]

As you go inwards, the arc length converges just as the integral of θe^θ from negative infinity to zero converges.

[@littlebigphil]

The arc length of a section decreases by a constant factor (1 over the ratio), so the geometric series describes the total length. Geometric series converges when the factor is less than 1, which it is because the sections are getting smaller.

[@Funkotronimus]

I’m gonna start a support group for Americans who pronounce “H” as “Haych,” and “Z” as “zed”

[@RiamiAurum]

Bob Trenwith that's the point, he's supporting those Americans tbat pronounce it that way

[@izicial7469]

I thought these guys were based in the UK. So wouldn't it make sense for them to say hache and zed???

[@ratlinggull2223]

As a foreigner, pronouncing h as 'eich' instead of 'heich' actually saves breath since your tongue isn't optimised for English. But Americans have no reason to because they're hecking native.

[@UnderscoreZeroLP]

Saying haytch isn't british

[@PhilBoswell]

+Underscore Zero it is when you want to read something over the phone and you don't want the recipient to think you're saying "eight". Yes, I know you can use the Phonetic Alphabet (which I learned almost before I could read ;-) but people are lazy :-P

[@hadhave7961]

This leaves me with so many more questions than answers

[@WillToWinvlog]

This video is PURE GOLD!!!

[@BiggieCheese]

Where my Platinum Ratio bois at?

[@ianmoore5502]

Platinum ratio gang rise up

[@user-mz7cn9hq8v]

Bruh platinum ratio is technically 1

[@yee6870]

@@user-mz7cn9hq8v iconic

[@sebastianzaczek]

Hey Numberphile! I recently was playing around with numbers and i came up with a rediculous fractal-like fraction (here is the first bit of it): ((((1/2)/(3/4))/((5/6)/(7/8)))/(((9/10)/(11/12))/((13/14)/(15/16)))) I hope you understand how it's built up. Then i wanted to see what this equals, and the larger i made the fraction, the closer it got to sqrt(2)/2: (1/2)=0.5 ((1/2)/(3/4))=0.666... (((1/2)/(3/4))/((5/6)/(7/8)))=0.7 ((((1/2)/(3/4)... (13/14)/(15/16)))) =0.7061728395... (((((1/2)/(3/4)... (29/30)/(31/32)))))=0.707023939... ((((((1/2)/(3/4)... (61/62)/(63/64))))))=0.7071021245... (I had to trick my calculator in a certain way to let me calculate this last equation, so the result might be slightly off) sqrt(2)/2 equals 0.7071067812... so the last result is equal for the first 5 digits after the decimal point. Now my question: If you continue this process infinitely, does the fraction actually converge towards sqrt(2)/2? And is there a way to prove it?

[@unclejoeoakland]

DerSibbe i think they call that a convergence...

[@mannyheffley9551]

but a fraction cannot be irrational so I think this assertion is incorrect

[@sebastianzaczek]

FReaKIng FReqUEncIEs i was thinking that too, on the other end however this fraction is theoretically infinite....

[@mannyheffley9551]

so then it is possibly irrational

[@sebastianzaczek]

FReaKIng FReqUEncIEs exactly... and there we start to need a proof... no idea how to proove/disproove it though...

[@georgecooper7389]

Extremely well edited!

[@davidcampos1463]

Thank you. I didn't know about these other ratios.

[@kujmous]

What is the ratio for 1, 1, 1, 3, 5, 9, 17, 31,… always adding the previous three values to get the next?

[@user-ct1ns6zw4z]

Kinda ugly ratio: 1/3(1+ cuberoot(19 - 3sqrt(33)) + cuberoot(19 + 3sqrt(33))) Which is about 1.84. Seems to converge pretty fast, 17*1.84 = 31.28 It's the root to this equation: r^3 - r^2 - r - 1 = 0 Because if we write it out in its recursive form: P_n = P_(n-1) + P_(n-2) + P_(n-3) Then divide to get the ratio: r = (P_n)/(P_(n-1)) = 1 + (P_(n-2))/(P_(n-1)) + (P_(n-3))/(P_(n-1)) We notice that as n->infinity, this equation tends to: r = 1 + 1/r + 1/r^2 Then we simply multiply by r^2 and bring everything to the other side.

[@IBioPoxI]

wouldn't it be 1, 1, 2, 4, 7, 13, 24 .... as ϵ+1+1 = 2 not 1 as you seem to suggest?

[@chrisroller1397]

Ben Fowler To me this is syraight out of /r/vxjunkies

[@kennethflorek8532]

Ben Fowler That series is the one that begins as 0, 1, 1, instead of the one that begins 1, 1, 1. If you need a number before 1, 1, 1, it is -1. That is: 1 = 1 + 1 + (-1)

[@AnonimityAssured]

A slightly more succinct representation is: _t_ = (1 + cbr(19 - √297) + cbr(19 + √297))/3. (cbr = cube root)

[@deanwinchest3906]

Don't forget to phile those nails when your finnished;;

[@briandiehl9257]

Why would he need to do that?

[@deanwinchest3906]

Brian Diehl thought it was mildly ironic to title/intro... Maybe a bit over the head✈️🐒

[@briandiehl9257]

I was thinking he could just turn the scissors when he is cutting and avoid all of this

[@deanwinchest3906]

Brian Diehl I prefer the old throw away dollar store *fingernail clippers* myself😄

[@ricardovalentin5056]

Fantastic! I love Numberphile!

[@vincentvanveen4436]

I'm so glad I clicked on this! Always had my doubts on nature sticking to one single ratio, seemed to simple.

[@mojoface]

I love reducing cognitive load. Probably my favorite thing actually.

[@joshpollack5936]

math is fun, adventurous, quirky, and clever. too bad it is delivered to us with the wonder completely striped

[@jpdemer5]

I like my wonder completely plaid.

[@swoondrones]

Great video!

[@p23570]

i learn so many new words watching these videos. hache, maths, etc...

[@st0ox]

Big in Japan lol

[@nowonmetube]

Savage

[@skyscraperfan]

Wouldn't it make more sense to define spirals somehow more continuosly, so that they are even self similar, if you rotate them any degree? They way you constructed them was just joining quarter circles together. In a real spiral there should not be parts of a circle anywhere. It should get smaller and smaller at any point.

[@xenontesla122]

warumbraucheichfüryoutubekommentareeinescheissgooglepluspagefragezeichen That type of portal is called a logarithmic spiral and it’s the type found in flight patterns and shell growth.

[@maxonmendel5757]

I love how comprehensive numberphile is

[@thesuomi8550]

0-bonacci sequence is my favourite

[@alphadad1966]

So European paper uses irrational values for its dimensions? A true A4 sheet of paper can never be accurately measured?

[@silkwesir1444]

The A4 standard is defined in terms of whole millimeters (210 × 297), and it has a tolerance of ±2 mm.

[@MsSlash89]

At least our sheets are, more or less, capable of keeping the same ratio when folded. Yours, once folded, just become another rectangle, not making any sense with all those "letter, legal..." comparing it to A2, A3, A4, A5...

[@OrcinusDrake]

A sheet with integer dimensions can never be exactly measured either

[@alphadad1966]

I should have said " A true A4 sheet of does not have dimensions that can be expressed in rational numbers"

[@blackhatguy6955]

No, the definition includes, "rounded to the nearest millimetre".

[@blue_tetris]

I never made it to the Silver Ratio without biting.

[@TofranBohk]

Mr. blue_tetris, how many spirals does it take to get to the SILVER ratio of a SILVER RATIO POP!?

[@JasmineDreams]

Why am I so excited about this!

[@SirDominic]

:O those t-shirts look amazing!

[@KnuxMaster368]

10th! hopefully it's not a Parker Square of a meme Edit: *sniff* I smell a Parker Square

[@ishaangovil5572]

The next is the bronze ratio...I think

[@sebastianzaczek]

MODERN SCIENCE i thought that too

[@hunnymonster2k]

Nah, the Parker Ratio ;-)

[@naverilllang]

Why should bronze, an alloy, come after two precious metals? Just saying. Platinum should have come next, ya know?

[@IncolasCopperfield]

thanks for a platinum video

[@johnsnow5305]

I've always loved geometry. It was my best math subject in school. When they started to introduce algebra and calculus and abstract trig (ie not showing how it actually plays out in physical space), it became less fun. I think it's important to combine the abstract facts we gain from geometry in an interesting way like you guys often do.

[@G8tr1522]

pretty much every great mathematician pre 1900 would agree with you I think.

[@zero56619]

Show me rubidium ratio

[@user-ct1ns6zw4z]

If gold is the 79th element and that gives you Sn = Sn-1 + Sn-1, and silver is the 47th element and that gives you Sn = Sn-1 + 2Sn-2, then for Rb = 37th element you could define it to be Sn = Sn-1 + 42/32Sn-2. The ratio for that one would be (42/32 + sqrt((42/32)^2 + 4))/2 = 1.8523537.... What am I doing with my life...

[@hanspeter9391]

Got me anxiety how he placed those scissors

[@testkiller6225]

the reason why we like geometric spirals in nature is the growth it represents, that it is alive. When we see it in art, it shows the artist's realization to make his art be alive.

[@markusjacobi-piepenbrink9795]

Just wonderful!

[@westsenkovec]

The silver ratio = *1:925*

[@RazvanMihaeanu]

Do that in Base 6! :)

[@ashishshukla8423]

I liked it earlier when animation was used only to show dynamic ideas which were difficult to describe on paper

[@NoNameAtAll2]

Ashish Shukla Agree It was a lot more interesting to watch the actual paper

[@iabervon]

Agreed, although I kind of like the animation for things that are trivial but not shown on the paper, such as adding two of these to one of those. If you muted the audio and just looked at the paper, he'd just be writing down some numbers, but the animation shows the calculation he's doing.

[@RickySTT]

The animations depict the concepts more accurately than the paper drawings do, though. (Mathematicians can’t necessarily draw straight.)

[@xevira]

The "Japanese" ratio could also be called the Electrum Ratio. You start off by removing a single square used to get a Golden Rectangle, and end up with a Silver Rectangle instead.

[@tifforo1]

Counting the fibonacci numbers backward into the negative: 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55