The Plastic Ratio - Numberphile

Ed Harriss discusses the plastic ratio - more amazing than the golden ratio? You decide!
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Ed Harriss is online at maxwelldemon.com and on Twitter at / gelada
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[@Bodyknock]

I believe an appropriate symbol for the Plastic Ratio would be ♻️

[@krisnaga91]

ahahaaaa you silly

[@tracyh5751]

+ marks for bringing triangles into the symbol.

[@alapikomamalolonui6424]

How about " פ "..? (Hebrew "Pe")

[@Krekkertje]

@@tracyh5751 not only that, but also a mobius strip

[@yimoawanardo]

I'd choose the arabic letter م Or ث

[@mebamme]

That triangle spiral is fascinating! Really didn't see that one coming.

[@CptStahlsworth]

mebamme I immediately find myself asking what it would look like as hexagons. How would that spiral look and what sorts of numbers would you find? Maybe I'll try it when I get home to pen and paper.

[@mebamme]

@@CptStahlsworth Interesting thought! I think the trouble with hexagons is that they don't line up like triangles and squares do, so you're not producing a new straight edge for a new tile to fit. Maybe the analogy works in different ways though?

[@Castle3179]

@@CptStahlsworth hexagons don't form larger lines as they are tesselated together. Maybe trapezoids or parallelograms?

[@nivolord]

You could also do it with rectangles, e.g. 1:2 ratio rectangles, aligning the longest side with the longest side. You get a weird sequence: 2, 2, 2, 3, 3.5, 4.75, .... given by a(n+2) = a(n+1)/2 + a(n).

[@sniceverything4944]

CptStahlsworth I don’t think you can it would just make a honeycomb pattern

[@Marconius6]

Can we get a more detailed explanation for how those calipers were designed? Like, what are the exact rations in it, how does it work out?

[@JasonDoege]

The exact(ish) ratio was exposed @3:09.

[@simonsays296]

I think that the rations of the calipers is just the plastic ratio, if you look at the thumbnail, you can see that if the top pivot is 90° it forms the two similar rectangles (bounded by the caliper arms) and a square formed by the tip of the third arm, up to the joint, over the cross bar, down the second arm to the joint (fourth side missing so it can move)

[@andykillsu]

There are no rations in the calipers

[@okktok]

Marconius yes, google is your friend

[@fuseteam]

and what ya do with it, never saw it in math class :p

[@captaincringe2595]

I don't think it's listed anywhere, but a few years back I found the plastic ratio, commonly denoted as ρ, has a very interesting primality-checking property. x is prime if mod(round(ρ^x),x)=0 So a rounded power of the plastic number will leave zero remainder when divided by its power only if it is prime (ignore trivial cases for x

[@vanhouten64]

nerd

[@MattMcIrvin]

Is that equivalent to the Perrin numbers? If so, it's *almost* true, but not quite. The first composite number that satisfies the test is 271441.

[@MattMcIrvin]

(*Note: the following is not quite right. See my subsequent comments.*) Answering my own question: It's not the Perrin numbers, it's the Padovan numbers. And it seems to be true at least up to 2000 or so. But in my experience these primality tests do tend to admit "pseudoprimes" eventually. For the Perrin numbers, *all* of the primes have the mod (P(x), x) = 0 property (it's a generalization of Fermat's Little Theorem), but there are very sparse composite numbers that satisfy it too, starting at 271441. They are called Perrin pseudoprimes.

[@supermarc]

As Matt McIrvin pointed out, it indeed unfortunately fails for x=271441=521^2, but the observation is very interesting (and by the way, it also works for 1

[@Peanutifyer]

vanhouten64 griff and breaden say hi

[@danielsieker9927]

A very interesting thing to notice is that the golden ratio, having a square in it, turns up in squares, while the plastic ratio, having a cube in it, turns up in cubes. The series of triangles is what you get if you do the "golden ratio-thing" with cubes in three dimensions and cut them all along the plane where the curve will lie in.

[@Lolwutdesu9000]

There wasn't much of an explanation as to where the x's come from. I feel as though there were a few points skipped over, making the video seem as though everything was poorly explained.

[@BenTheSkipper]

He decided to call it a mystery number so that he can explain the ratio

[@BenTheSkipper]

Correct me if I'm wrong

[@applesarefum]

@@BenTheSkipper I think they meant that the video doesn't explain why the third section is x², the first two sum to x³, and so forth

[@AaronHollander314]

He says that x^2 equates to the (x+1), but then calls it x^3...I am lost...lol...and it's not even clear that the x^2 is accurate.

[@EebstertheGreat]

Yeah, the details of the scaling were left out. The idea is that if you dilate one set of calipers until one of its gaps lines up with a larger gap in the other, you end up also lining up the other gaps. So dilating by x (to make the length 1 gap line up with the length x gap) is the same as adding two smaller gaps.

[@theninjafroot]

Oh hey, Ed Harris was my calc 3 professor! He had amazing lectures and I got a much deeper understanding of calculus in that class

[@bobbun9630]

i managed to get through the math program where he teaches never having him as an instructor, though a number of classmates who had him for Intro to Proof spoke very highly of him. My only contact was once at a math club meeting and once when he subbed for my normal instructor in Abstract Linear Algebra.

[@norrisleung666]

How can u get deeper understanding instead of confusion?

[@Hecatonicosachoron]

I do spy the little trick with the cubes, I do look forward to seeing that :) There's also a group of recursive polynomials with the form X^p = X + k ...they are interesting to cover.

[@Triantalex]

??.

[@Frownlandia]

This is one of my favorite numbers; I've been waiting for this video for a while. He didn't say it this way, but whereas the golden ratio is (a+b)/a=a/b, the plastic number is (a+b+c)/(a+b)=(a+b)/(b+c)=(b+c)/a=a/b=b/c. And there isn't a way to do that with a higher-degree algebraic number! The plastic ratio really is the best you can do. There's also a spiral that can be made of cubes with side lengths of the Padovan numbers. The other conspicuously missing fact is that analogous to the golden ratio, there is an infinite nested radical expression for psi which doesn't work very well in plaintext, but it's the cube root of 1 plus the cube root of 1 plus the cube root of 1...=psi.

[@chrisg3030]

Also the golden ratio is x -> y, y -> xy. Thus x, y, xy, yxy, xyyxy, yxyxyyxy . . . The plastic number is x -> y, y -> z, z -> xy.

[@benjaminnewlon7865]

I would just play with the dividers. No math, just bending them.

[@BenTheSkipper]

That is the beauty of mathematics... There's an infinite number of problems to solve and enjoy

[@BenTheSkipper]

@Crockett so what is your interpretation of what he was saying?

[@jojololo9157]

What are these thing? What do they divide? Sorry im a country boy, i dont understand what these things are for...

[@nowonmetube]

@Crockett I know right? Like wtf!

[@nowonmetube]

@@BenTheSkipper interpretation? He's just saying that he'd just play with them! As with a toy for stress relief or something.

[@Rhynez]

Hmm. This video is harder to understand. You guys usually ease into the matter and give an example and then slowly work it out. But here you start with these weird dividers and I have no idea what they are doing.

[@muchozolf]

Henry B Yeah, it's kinda funny. The video begins and suddenly you have no idea what are you watching.

[@shannu_boi]

Yea same

[@sailor5853]

Learn math then

[@LordHondros]

I know this video was released a while ago, but I absolutely love these types of ratios. Ed, or Numberphile in general, you may find this interesting (or honestly probably already know about it, let's not kid myself). The golden ratio can be generalized to a form f(x) = x^2 - ox - 1 , where o is 1, and the [positive] solution is the golden ratio, phi, ~1.618. Similarily, we can generalize the formula for the plastic ratio to f(x) = x^3 - ox - 1, where o is 1, and the [positive, real] solution is the plastic ratio, ~1.325. Something very interesting happens if we set o to 2 in the plastic ratio formula. If we solve for x, the [positive] solution ends up becoming the golden ratio. More fun things: if we set o to 3.5 in the plastic formula, the positive solution is 2. What fun! Unfortunately I was unable to get the silver ratio in a nice pretty number in the plastic formula, and I'm also unsure if any other metal ratio ends up showing up in a nice number either. I will continue messing around with this new formula however, I love these seemingly random ratios!

[@jdavis.fw303]

Now I'm super curious about the math behind why you can't do it with 5 or more "pins". That seems significant.

[@PhilippeAnton]

Me too, especially considering the fact that the same thing with 5 pins, a ratio around 1.220744085 (solution of x^4=x+1, ratio of the series u(n)=u(n-3)+u(n-4) ) seems to have all the properties illustrated at the beginning of the video. Of course, I don't think there is an elegant spiral to build with this ratio, but is that the point?

[@zachariahkindle8926]

@@PhilippeAnton no I dont think so. I think the issue is that ratio wont hold out farther, but I'm not sure. I'd like to see them discuss it as well

[@PhilippeAnton]

@@zachariahkindle8926 Can you elaborate the "not holding out farther" part? As far as I can tell, the mechanism seems to be exactly the same, and the recursive relationship won't suddenly stop being true after some steps.

[@zachariahkindle8926]

@@PhilippeAnton sure. I just ment that it does in fact work for x^4 but I dont think it builds up the same way for the powers higher. Honestly I'm not 100% it's been a few years since I covered this in a class, but when I get a break later I will try to look into it. You may in fact be correct, I was just fairly sure the math fell apart on higher powers. I'll try to get back to you in the next 24-48 hours

[@ChrisLuigiTails]

3 pins --> golden ratio --> squares 4 pins --> plastic ratio --> triangles 5 pins --> another ratio --> lines? And you can't make a spiral with that I don't have any "proof" or whatever but MAYBE That would mean that we'd have pentagons for 2 pins but you can't really do many things with 2 pins

[@maitland1007]

Great subject. But the explanation seemed a bit unorganized and hard to follow.. skipping steps, etc.

[@ceselb]

Have to agree on this. Not well explained at all.

[@theprogrammer32]

I know right, like how do the calipers work, how were they made, and more explanation for each number he writes on the page, some of them seemed out of nowhere and there was no explanation.

[@vsevolodmakarov9067]

It was really greate and interesting,but I can’t say that I understand everything

[@matthewcapobianco9332]

Yeah he wasn't really explaining anything he was doing. Some more setup would have helped out a lot.

[@MattMcIrvin]

He didn't quite explain that the ratio of successive numbers in the Padovan sequence *converges* to the plastic number as they get large, just as the ratio of successive Fibonacci numbers converges to the golden ratio. Numbers of this general sort are called Pisot-Vijayaraghavan numbers, or PV numbers. If I recall correctly, the plastic number is the *smallest* PV number. For every one of them, there is a whole family of Fibonacci-like integer sequences whose ratios converge to it. Another one for the plastic ratio is called the Perrin numbers, which begin differently and then iterate the same way as the Padovan sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29... This sequence is interesting for inducing a remarkable probable-prime-number test, but that is another story.

[@skwisgaarskwigelf5381]

I doubt many other people could make that claim either.

[@hubson7416]

It was really greate and interesting,but I can’t say that I understand anything

[@Triantalex]

false.

[@iowain8623]

"If x^3=x+1 then we can just multiply x and 1. So THIS is going to be x^4." WHAT?

[@bumpsy]

the ratio is x. Thus every gap is exactly (the gap before)*x. First gap is 1 long. Second is 1*x=x. Third gap is x*x=x^2. Fourth x^2*x=x^3 and so on

[@bumpsy]

@@bosstowndynamics5488 well, reading the comments, almost every question here that came from "he didn't explain that" can easily be answered from the video. I agree that the structure could be better but he doesn't have to do ALL the work for you. Most people here just don't seem to want to think a bit on their own. You can also always pause and recapitulate after every step. Also I don't think he left anything out really. The title says "ratio" and in the beginning he shows the caliper and it's gaps and how they change at an equal rate. For me it seems just a normal logical visualisation of any ratio. (There's a caliper for the golden ratio as well for example)

[@V8Murder]

​@@bumpsy Ah... x is representing the ratio. Gotcha.

[@alex.mojaki]

What about that 3D box construction in the background? I was waiting to hear about it. I assume it has to do with the 3 in x = x^3 + 1.

[@OlafDoschke]

To me it looked like a 3d of the golden ratio squares, just with cubes, but yes, that was sitting there and the context is missing. Victim of editing, I assume.

[@niekpauwels9569]

Maybe there will be an extra video where they explain what that does too.

[@ceruchi2084]

Wait six months for the next video from this interview haha.

[@iowain8623]

Should we give the points names like a, b, c and d? Nah, I'll just call them here, here here and here. That won't be confusing at all.

[@CharmEng89]

yeah i felt this would also have been useful...

[@KnakuanaRka]

Yeah, making them distinct would be easier to follow.

[@quarkonium3795]

I kinda wish he hadn't done the divider thing at all. Just start with the triangle spiral, it's much easier to follow

[@dyllpickalio1700]

Next video: *The Bacon Ratio*

[@michella1913]

This episode was truly a great one. Thank you Numberphile!

[@WBvanKempen]

In the plastic ratio it's not only the sum of the second and third position before, but also from upon position 6 it's the sum of the first and fifth position before. So: Position 6: 1+2 (3th + 4th), and 1+2 (1st + 5th) Position 7: 2+2 (4th and 5th), and 1+3 (2nd + 6th) Position 8: 2+3, and 1+4 Position 9: 3+4, and 2+5 Position 10: 4+5, and 2+7 Etc. This is also shown in the drawing: the length of the new tiangle is the sum of the triangle before, and the triangle 4 steps before that one (so position 1 and 5 before). This only works with a start from position 6 because of the chosen start with 1, 1, 1. That is just an agreement, not a fact. Also: in the golden ratio you could go backwards though position 1 and it still kind of works. You will get the same numbers, but for every even negative position it will be the negative 'brother'. In de plastic ratio you can't go backwards like that in a 'normal' way. The numbers in the negative positions will go everywhere.

[@centerbfd]

And more. Not only can you add two in series to get the fourth, but you can add three or five in series to get the sixth or ninth, respectively. Now _that's_ cool!

[@MelodiCat753]

Cool topic, but the explanation was too fast and confusing.

[@wcsxwcsx]

I sometimes reduce the speed on videos like these. It can be very helpful.

[@randomdude9135]

Yup. I didn't get anything, so I got bored, n now I'm just seeing comments :/

[@harryxiro]

I just discovered something crazy about the plastic ratio! The Golden Ratio squared equals itself plus one. The Plastic Ratio cubed equals itself plus 1. Mind blowing!

[@danielhmorgan]

nobody ever mentions that. Thanks!

[@sageinit]

FINALLY. THANK YOU SO MUCH FOR FINALLY MAKING THIS VIDEO.

[@danielsteel5251]

Thanks to patrons, the video ends with calming piano music, instead of a weird, unrelated advertisement.

[@BenTheSkipper]

Yes, I also noticed the same thing

[@OpinionAxe]

This was a very confusing video if you have no freakin' idea what those red plastic thingies are.. how they work.. and why one can write x-squared for the distance between the 3rd and 4th leg. It would be so helpful if another short video was added why these math rules are actually valid for this red plastic thing.

[@danielf.7151]

"and why one can write x-squared for the distance between the 3rd and 4th leg" If i understood it correctly, that's simply how they defined it. The problem with the video is that he didn't give the parts any names, just calling them this, this and that, which makes comunication a bit difficult. We have 4 Points: A, B, C, D. they are chisen in such a way that the ratio between BC and AB is the same as CD and BC. We call this ratio X. We say that AB has length 1. Therefore, BC = (AB)1 * X(ratio BC/AB) = X Then, CD = BC * X = X * X = X². I think the confusion comes from the fact that X represents both a ratio and a length.

[@chrisg3030]

@@danielf.7151 But X represents both a ratio and a length in the case of the Golden Ratio too. Imagine the calipers have 3 points ABC chosen (or linked) in such a way that the ratio between AC and BC is the same as that between BC and AB. We say AB has length 1, and put what we know to be the number on the ratio, approx 1.618. This is also the length BC, and the length AC is 1.618+1. And 2.618/1.618 ≈ 1.618/1. I think what could be particularly confusing about the plastic ratio is we not only have X^3 = X^1 + 1, but also X^5 = X^4 + 1. Please check all this.

[@ActuatedGear]

I really like the flow in the infographic. Nice ki blast.

[@sumdumbmick]

@3:20 the material called 'plastic' is named that because it is described by the term. the meaning did not change.

[@MattMcIrvin]

My impression was that "plastic" in that context originally referred to "the plastic arts", e. g. sculpture and the making of objects. Which is pretty much where the material gets its name too.

[@sumdumbmick]

All uses of 'plastic', ever, indicate the meaning is simply 'deformable'. Any sense you might have that it means something more is coming from failing to recognize that you've wrongly incorporated extra details from a specific context into your espoused definition.

[@JavedAlam24]

@@sumdumbmick The term has been colloquially and literally associated with the material plastic, and is used less frequently to refer to the concept in contemporary usage.

[@moskthinks9801]

Implies also that x^5=x^2+x+1, which of course can be factorized.

[@danielf.7151]

IMO that would have been the more intuitive step. Why did he do the multiplication? I mean, it works, but to me, it felt like he was working backwards.

[@M4rtingale]

I did not get how he got from x to concluding the next space was equal to x squared. It’s clearly not x squared. If the first space was 5cm, the second was certainly not 25cm. It might be late, but where am I going wrong here?

[@hedaurabesh]

X is not the length, but the ratio between it and the first unit-distance. So if the first space was of length 1, the next space is of length (1*X), where X is the plastic ratio. Then the next one is ((1*X)*X) or -- X^2. And so on. So if the second unit was your 5 cm, then the first unit is (5/1.324...), and the third unit (the one you thought to be 25cm) is actually (5*1.324..), or about 6.62 -- which closely matches the calipers in the video, it seems. Where the X^2 comes from is that that is its ratio to the first unit distance: (5/1.324)*(1.324)^2 = (5*1.324). Only if you set your first caliper distance to 1, will the distances be X, X^2, X^3, etc. On any other distance it is the ratio, not the actual distance.

[@yaeldillies]

It's because the spaces between arms of the calliper are *proportional* to 1, x, x^2... You may have thought about this noticing that cm^2 aren't really a length.

[@robbrown9879]

It might help to keep in mind that when squaring numbers close to 1, the absolute increase isn't that big. We aren't going from 5 to 25, but rather from about 1.3 to about 1.7 = 1.3*1.3. Also, as others have noted, we chose the first gap to be 1 (no unit). If this 1 represents 5cm, then x represents x*5 cm and then x^2 represents (x^2)*5 cm (so around 1.7*5 = 8.5cm).

[@M4rtingale]

Thx guys, I get it now! Much appreciated and thx very much for your very thorough answers!

[@maitland1007]

I love how your videos often explore stuff like 'what would happen if we generalized this or extended it further'. It would be fun to see another video that did this with this ratio stuff (other shapes, exponents etc).

[@flamingpaper7751]

I love learning all these ratios

[@SteinGauslaaStrindhaug]

Is it a coincidence that this ratio is _very_ close to the ratio 4/3 as in the standard 4:3 aspect ratio? Or was 4:3 picked deliberately as a simple ratio approximating the plastic ratio, or just because that ratio appeals to humans...

[@ScarfmonsterWR]

If I remember correctly, Thomas Edison is basically who picked and promoted the ratio first with his 35mm motion picture films. I tried to find out why exactly he chose that ratio but couldn't find anything beyond some anecdotal mention that his lab equipment at the time used similar ratios.

[@eractess]

Based on the fact that fewer and fewer screens are 4:3 these days I would argue that it isn't that appealing. It's just more convenient for glass tubes.

[@coopergates9680]

@@eractess No common aspect ratios are nearer to the golden ratio than 8:5? I have not seen 13:8 used.

[@walkingwriter4325]

Padovan Numbers? Who knew? Awesome!

[@yuryschkatula9026]

Jedi numbers for sure!

[@Dalenthas]

I think it's kinda buried in here, but I want to point out that 1 is x^0, so while you can write it as x^3=x+1, you can also write it as x^3=x^1+x^0.

[@topsecret1837]

That would be abstracting it though

[@Dalenthas]

@@topsecret1837 true, but it also reveals an otherwise hidden structure to the calipers comparison.

[@MattMcIrvin]

IF you assume that x^0=1 even for x=0. Which, when mathematicians are talking about polynomials, they usually do (even if your math teacher doesn't agree).

[@Dalenthas]

@@MattMcIrvin if x=0 the whole dammed thing breaks down. It becomes 0 = 0+1, which is nonsensical.

[@bumpsy]

but why would you write it as x^3=x^1+x^0? If the first gap wasn't 1,this wouldn't make sense anymore also I think X is just a name for the ratio itself here (which is around 1.3247)

[@EpicMathTime]

Hmm, can't do it with 5 prongs? is that related to the unsolvability of the quintic?

[@pianojay5146]

interesting

[@gjugany]

very

[@sundeco7467]

Doubt it. It shouldn't matter if the polynomial is unsolvable over radicals (which is what you're referring to), as long as there is a real solution, that solution can be the ratio. Edit: It can't work if you want all the distances to be different: Say you have the five prongs in a line in the order A,B,C,D,E. Then AB=1, BC=x, CD=x^2, DE=x^3. The polynomial would be AC=1+x=x^4. Then you also have the distances AD,BD,BE,CE which must be x to the power 5,6,7,8, in some order. Then AE=x^9 which you can verify does not equal 1+x+x^2+x^3.

[@pianojay5146]

@@sundeco7467 Oh I just noticed now and tried to comment that same thing. Everything seems fine.

[@shirou9790]

Furthermore the polynomial would be of order 4.

[@flamingpaper7751]

For those curious the exact value of the plastic ratio is Cube root((9 + square root(69))/18) + Cube root((9 - squate root(69))/18)

[@TruthNerds]

Alternatively, with w=cube root((1+square root(23/27))/2), the ratio is w+1/(3w).

[@ScormGaming]

I am going to look at it in its written form to see how beautiful it looks.

[@Hecatonicosachoron]

There's also a problem for anyone interested: Given any recurrence relation, any integer initial conditions and the characteristic polynomial for that recursive relation, IF the characteristic polynomial has MULTIPLE real roots, THEN WHICH ROOT will the ratio of successive terms in the recurrence sequence converge to? So sorry for posting so many comments, I just love that subject.

[@Hecatonicosachoron]

Sebastian Henkins yes, they are the attractive fixed points, but there can be multiple and different ones result from different initial conditions. Some initial conditions do not converge but fluctuate.

[@Hecatonicosachoron]

Sebastian Henkins essentially we are partitioning the space of initial conditions to the roots that they converge to. Sounds like some fractal patterns will be possible by extending the IC to complex numbers (then the solution is a hypersurface of that object) Numerical methods are well-explored, but I was wondering if there is a general analytical result.

[@roqxwalker3896]

I’m glad you finally covered the Plastic Number

[@liamsmith4017]

I apologize if I'm not the first to mention, but: There are three ways to divide a square into three similar rectangles. One is to divide it into three 3:1 rectangles of equal size. Another would be three 3:2 rectangles, one twice the size of the others. The third is three rectangles of proportion x^2:1, where the ratio of smallest to middle is x^2, middle to largest is x, so smallest to largest is x^3. There's a lovely image on Wikipedia, and the construction takes advantage of the fact that x^5 = x^3 + x^2 = x^4 + 1.

[@OlafDoschke]

Fantastic how you come up with these things over and over again. Never heard of this and I agree with Ed Harris, there should be more to find about such geometric ratios. You already had the other series resembling other spirals (silver ratio), but no, I never heard of this before.

[@erikheddergott5514]

Ian Stewart calls the Padua Number Sienna Number, since Fibonnaci was from Pisa.

[@jacemcpherson]

Dr. Harris!!! Long time no see! Never thought the next time I'd see you would be on a Numberphile video.

[@wktodd]

Arrggh I was doing something like this this morning (and struggling ) trying to work out the length of an expanding coil!

[@A1rPun]

I'd love to see your work. I'm trying to create a padovan visualizer myself after this vid

[@crushelnast6657]

Everybody gangsta till a spherical ratio rolls in

[@Tondadrd]

x**3 = x + 1 I came to this equation during a test on derivatives about a year ago and as I couldn't solve exactly and seemed to have missed something, I approximated it. It was one of the inflex points and I needed it for the graph of the function it was derivative of... When I asked my teacher afterwards whether my working out was right, he replied that he made a mistake in the test assignment and made sure I didn't get stuck on that one thing and that I made the other parts of the test as well. And when I asked about the number, he replied he doesn't know it to more decimal places, which made me feel like I was actually discovering something about this number for my self during the test. It is quite a memory as I were sitting there scribing and approximating the number to about four decimal places, trying few irrationals including pi, whether it is half of it or something (I knew that pi should have no business here, but as I said, I seemed to have forgotten something or didn't spot something obvious.) Well anyway. Good video with plenty of time for the viewer to stop at any time before any question to try to ponder or flat out come up with the answer themselves!

[@dialecticalmonist3405]

I say this about once every Pi videos, but this is my new favorite thing in math.

[@ChongFrisbee]

Anyone else eagerly anticipating the video with the transparent boxes featured in the back?

[@pafnutiytheartist]

Can you show why there is no way of setting it up with five pins?

[@JordanMetroidManiac]

That’s because the caliper is assumed to be 2D. In 3D, a caliper with such properties have have up to six pins, but not seven or more.

[@mrdr9534]

@@JordanMetroidManiac Though Your answer is correct as far as I understand it, I don't quite think that it really properly "explains" why that is the case. And unfortunately I don't think I'm up to the task either, as it (if I'm not mistaken) requires You to explain in words how, the "hinges" of the fifth pin wouldn't be possible to position "correctly" AND still to work i a 2-d plane. The third pin (and consequently it's "hinges") would have to have the ability to move in the "3'rd dimension" otherwise it would "be bound" in a single position, dictated by the positions given from the hinges in the other "preceding" pins... Yea I can't really explain it either, though it might possibly be due to me I having misconstrued how the whole thing works, or rather doesn't work in this case :) Best regards.

[@theleastcreative]

That was super refreshing

[@themorgenix4848]

Felt like I came into a video that had already been going on for 15 minutes and I couldn't really follow

[@jmm616]

TheMorgenix exactly... there felt like there was so much setup missing

[@themorgenix4848]

@@jmm616 Someone told me in comments that x is basically what you multiply the previous length by to get the next length, so after 1 it's 1*x = x, and then x*x = x^2

[@oliverpackham6278]

I don't know why, but I'm super interested in this.

[@srmendoza]

I saw a cube on the desk that was never address, please elaborate on it next time.

[@afaegfsgsdef]

Why is the whole thing x^2 Times x+1 instead of x^2 PLUS x+1?

[@EpicMathTime]

Because x^3 = x+1.

[@TonyboyDK]

It's more intuitive if we do the calculations in the opposite direction. We want to find an expression for x^5, and we know that x^3 = x + 1, then x^5 = x^2 * x^3 = x^2 * (x + 1) You are, however, correct that this length is ALSO x^2 + x + 1. This is why, when we solve for x in the equation x^2 + x + 1 = x^2 * (x + 1), we get x = 1.3247... The plastic number :D

[@zeldajerk]

Because they are equal. (x^2) + (x+1) = (x^2)(x+1) Why? We are given that (x+1)=(x^3). so (x^2)+(x+1)=(x^3)+(x^2)=(x^2)x + (x^2)=(x^2)(x+1) with factor by grouping.

[@gissehel]

Because who cares, addition, multiplication, it's all the same... Of course it's not. It just appear that HERE, (x^2)(x+1) = (x^2)+(x+1). In fact, the plastic ratio is the only real number that has this property. Let's see it. (x^2) + (x+1) = x^2 + x^3 (remember, x^3 = x+1) (x^2) + (x+1) = x^2 (x+1) (factorize x^2) And that's it, already there. They are equal !

[@chrisg3030]

One possible source of confusion became clearer to me on comparing the three ratios: the Golden or Φ ( approx 1.6180), the Plastic Ѱ (1.3247), and the next in the series - call it the Paper or P - at 1.2207. The Golden derives from the Fibonacci sequence recursion of An = An-1 + An-2, the Plastic from the Padovan An = An-2 + An-3, and the Paper from An = An-3 +An-4. Now Φ^2 = Φ+1, Ѱ^3 = Ѱ+1, and P^4 = P+1. But what's peculiar about Ѱ is that Ѱ^5 = Ѱ^4+1 as well, hence the unexpected result of those calipers.

[@MarceldeJong]

Why is the length of that last section of the caliper x^2? That one I didn't understand

[@bumpsy]

So idk if you still care but that's how I understood it: The first gap (or section) was 1 unit long (It could be anything but it's easier with 1). The ratio at which those gaps get bigger ("plastic ratio", of course) he names X (just a place holder for 1.3247). So, every gap is exactly X times bigger than the gap before. First gap = 1 Second gap = X*(First gap) = X*1 = 1 Third gap (your question) = X*(Second gap) = X*X = X^2

[@madskjeldsen9593]

Does this provide a geometric way of doing cube roots? Nice video as always!

[@Xayuap]

of course not, that is already proven to not be feasible in 2d

[@TheDentrassi]

Huh. I love seeing stuff like this. I am very visual and struggle with maths from a numbers perspective. Start throwing in geometry and it makes so much more sense. Can anyone recommend anything on geometry for artists that teaches maths in a really visual way. The plastic ratio feels very plant like.

[@JwalinBhatt]

Fun fact, it is conjectured that: plastic ratio is the smallest number between 1 and 2 whose powers will give near integers. And golden ratio is the largest number between 1 and 2 whose powers will give near integers.

[@AssassinGrudge]

how is it possible to start from a triangle and up with more sides than a rectangle

[@marlou169]

Google pictures of the interior or better visit the abbey Benedictusberg, because the spatial experience of sitting, walking and being in this proportional system of the ‘Plastic Number’ is quite unique!

[@williampotter1004]

At 11:34 I think reason that you cant make the 5 prong caliper have to do with that the general quintic is unsolvable.

[@Hecatonicosachoron]

My favourite algebraic constant!!!

[@gregoryzy]

Okay, I already comented on a older video o numberphile, but will do it again because this video got closer to what I was looking for, that is, a sequence that follows the pattern "add the three previous numbers to ge the next", for example, [0, 1, 1, 2, 4, 7, 13, 24, 44, 81...]; and the ration between the numbers (eg.: 81/44) gets closer and closer to "1.8392..." but I don't know where I can find a formula like the Fibonacci's. So far, I only know this: X = [ A(n-2) + A(n-1) + A(n) ] / A(n) Note: [ A(n-2) + A(n-1) + A(n) ] = A(n+1)

[@TrueMathSquare]

What happens if you do it with regular hexagon? Is it possible? It's the last regular polygon that does tessellate. Also what about polygons that aren't regular that still does tessellate like a pentagon that is shape like an house? What effect would that has?

[@robbrown9879]

As the angles of the corners of a hexagon don't divide 180 degrees, like a square (2x90) or triangle (3x60) do, you don't get a larger flat edge when you start joining hexagons in a spiral. So you just put 3 hexagons together and you've completed the circle (rather than a spiral). I guess you could say the ratio you end up with is 1.

[@TrueMathSquare]

@@robbrown9879 How about an shape like a pentagon that is irregular?

[@sumantchopde9039]

Congratulations to Ed Harriss for getting the role of Angel in Money Heist! XD

[@flowerwithamachinegun2692]

When will we get the bronze ratio? Or the synthetic ratio? Or the food ratio????

[@andykillsu]

Where was the Pi day video??? Kinda saddened that you stopped that tradition this year 😦

[@nowonmetube]

Oh boy...

[@dmytromalevskyi400]

this was very cool!!

[@rosiefay7283]

0:06 "the jump from here we have the jump from this one to this, and that's the same on this set from here to here". Lots of jumps and thises. "The second division grows at the same rate." Huh? 0:32 "you can work out what the number is" What what number is?? I had no idea we were supposed to answer a question. Sorry, but you're not making yourself clear at all. :(

[@chetseidel]

This is the worst Numberphile video I have ever seen.

[@raykent3211]

Feed the natural numbers into this machine and it spews out alternating digits of pi and phi. I won't tell you how it does that.... Disappointing for someone who wants to know how a machine works.

[@radadadadee]

I had the same reaction. And yet, some people in the comments say it's the best video of numberphile?? I don't get it. It seems like I was watching the second part of a video for which I miss the first part. I was like WTF is that caliper?

[@tildessmoo]

I realize that it's really two different explanations for the same ratio, but the way you described it, it looks like you're saying that 1³ = 2. Also, while I'm not generally one to nitpick using a random symbol to represent a constant for a one-off explanation, since you're deliberately avoiding φ (phi) because it's used for the golden ratio, ψ (psi) really isn't the best replacement, since it's used for the supergolden ratio. You might have been better off using a different alphabet (פּ, the Hebrew letter "pe," maybe?) or doing a quick Google search for the plastic ratio before filming, because it says right at the top of the Wikipedia page for plastic number that the symbol for the ratio is ρ (rho).

[@gurrrn1102]

Rho - it has a long Wikipedia page but I was looking for a way these sequences can be extended for the sums of arbitrarily many prior terms arbitrarily far back in the sequence. Some extension of Lucas sequences?

[@tom_something]

American here. The geometric relationship at 8:43 ... that's the same as the A-sized paper you guys use, right? A3, A4, etc.

[@danielf.7151]

No. With the A-sizes, the ration between the long and the short side is always the square root of 2. That way, you can cut it in half and maintain the ratio.

[@tom_something]

@@danielf.7151 whoops, yeah. Thanks.

[@Catman_321]

yo so recently i was screwing around with wolframalpha and desmos to try and make numbers similar to the metallic ratios but instead make n^3=mn+1 and also coincidentally have the property 1/x+m/x^2=x and i have determined it to be f(m)=cbrt(1/2+sqrt(1/4-(m/3)^3)+cbrt(1/2-sqrt(1/4-(m/3)^3) this includes the plastic ratio when m=1, interestingly enough the golden ratio for m=2, and other "petroleum ratios" (i invented that name while writing this lol) for m= any value, which will in turn equal f(m)^3=mf(m)+1

[@LeoStaley]

I need more on this. Please.

[@toomdog]

At 1:40, when he's first laying out values for the divisions, how can he just say that the third space is x^2, and the fourth distance is x^3 and so on? when he writes his x's below the sequence at the end, it works because x happens to be one, but it doesn't look like the x^2 distance in the beginning is equal to the x distance. Am I just being picky about dividers and missing the point here? It seems like he just said, "I'm going to call this x^2," as if that were another independent variable.

[@dutchdykefinger]

7:23 i'd argue the squares have more sides (4 vs 3), but you interpret the axes, which are 2, where it is 3 for the triangles, and 2 for the square. i know, semantics, and i know what was meant, but just for clarity.

[@BIZEB]

I think he meant that the polygon built from the triangles have more sides than what would be built by the squares (which is a rectangle). No?

[@cr1216]

There's one thing though. Indeed 3+4 is 7 and 4+5 is 9 but from the picture we get 7 from 2+5 and 9 from 2+7. It's clear a[n]= a[n-1]+a[n-5] but it need to be proven that a[n] = a[n-2]+a[n-3]. I suppose it can be proven by induction using the first several number as base case that a[n-2]+a[n-3]=a[n-1]+a[n-5] but it does not follow directly from the picture.

[@JJ-kl7eq]

The Plastic Ratio is also 1.58577251. The ratio of length to width of most credit cards.

[@dlevi67]

For a cubic, use Cardano's formula. For everything else, there's MasterCard.

[@SirNobleIZH]

Why can't the symbol be capital phi, to show superiority over the golden ratio (lowercase phi)?

[@niekpauwels9569]

Could the 5 pin divider be done with a 3D construction or does the dimension in which you make dividers not matter?

[@niekpauwels9569]

Obviously a similar sequence can be made by skipping 3 numbers every time: 1 1 1 1 2 2 2 3 4 4 5 7 8 9 12 15 17 21 27 32 38 48 59 70 86 107 ... (if I've done this right)

[@oogrooq]

Too bad there's no January 32nd.

[@user-mt9ux2di6u]

Is there a ratio for every material? Does a thoungsten(I don't know how to spell that) ratio exist?

[@aviralshukla3009]

Love numberphile...💓💓💓💓 Love you Brady...💓💓

[@sturniboy]

This video was great.

[@vikramb5733]

First time i watched carelessly. Second time i watched 💡 wow😍

[@Axacqk]

The last bit about there being no "wooden ratio", "clay ratio" etc. is the main takeaway. I really thought there was an infinite sequence of higher degree ratios.

[@makhnoboi1996]

There are the mettalic ratios stemming from the golden ratio, I wonder if there are different "polymer ratios" stemming from the plastic ratio

[@TheBeatle49]

I love Numberphile but I don't follow this. Any help here?

[@asdfasdf-dd9lk]

Basically the ratio, x, is the value you multiply a length by to get to the next length. So the length after your first length (if you assume the first length's one), is x, the ratio of that one is x^2, the ratio of the one after is x^3, et cetera.

[@johnbennett1465]

@@asdfasdf-dd9lk - So the first three elements are 1 1 1 and they are also 1 X X^2. Thus X is 1 and thus the series is 1 1 1 1 1 ..., oops somehow X^3 is 2?!?! I still have no idea how addition is being turned into multiplication.

[@PhilBagels]

@@johnbennett1465 The Padovan sequence is just the integers you get making successive approximations of the Plastic Ratio. Just as the Fibonacci sequence gives successive approximations of the Golden Ratio. 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, etc., gets closer and closer to the Golden Ratio. 1/1, 2/1, 2/2, 3/2, 4/3, 5/4, 7/5, 9/7, 12/9, 16/12, 21/16, 28/21, 37/28, 49/37, 65/49, etc., gets closer and closer to the Plastic Ratio.

[@johnbennett1465]

@@PhilBagels - Ok, thanks. The video does not make this clear.

[@PhilBagels]

Did Numberphile ever make a video about the Tribonacci Sequence? You can probably guess what that means: you add up the previous three numbers to get the next one: 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, etc. (For some reason, they usually start it with 1, 1, 2 [or 0, 1, 1] instead of 1, 1, 1. I guess it just works out a little "smoother" that way.) And there's also the "Narayana's Cows" sequence, where you add the previous and the third previous to get the next number: 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, ... And there are plenty of other, more complicated sequences, that all have their associated constants. I suppose Numberphile should eventually do videos on these other ones (they don't have to do all of them - the more complicated ones become less interesting and less useful.) Here's one more sequence, can you figure out the rule: 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, ... This sequence also has a name, but I forgot what it's called.

[@wolf1066]

Thanks to this vid, there are now two sets of calipers out there that I want to buy...

[@nondamaroulis4342]

Is this the ratio used for the iso paper sizes? A0, A1, A2, A3 etc?

[@reamick]

No. The ratio of the sides in ISO paper sizes is sqrt(2), so that cutting an A2 sheet in half crosswise will produce two A3's.

[@SolidSnake123321]

Anyone know the music in the ending? It's amazingly calming.

[@kurzackd]

2:55 - should be x^2 PLUS x^3 no?

[@EpicMathTime]

It's both. Recall that x^3=x+1, so x^3+x^2=x^2(x+1) = x^2(x^3). He just didn't explain what he was doing very well.

[@sagov9]

this particalur x has the property that x^3+x^2 = x^3*x^2, just take the formula x^3 = x+1 and multiply by x^2 on both sides. But I agree, x^2*(x+1) came out of nowhere, and the derivation of the total length should have been: total length = x^2+x+1 = x^2+x^3 = x^2(1+x) = x^2*x^3 = x^5. Mixing up the operator kind of let him start at the third step.

[@Horforia]

When he says at the end that there is no way to do the calipers with 5 points, does he mean that there is no "known" way, or that it's been proven that it can't be done?

[@Alexander-oh8ry]

What happens when you do the drawing thing with hexagons?

[@edmundharriss2288]

Have a go,I have not been able to make anything meaningful, but there might be something!

[@DerBard2789]

If you're having trouble understanding this (like I was), I recommend watching 1:40 - 2:00, and then watching 0:05 - 0:20 (back and forth a couple times maybe). These are the two crucial points you need to understand as a foundation before the "building" done in the rest of the video. (1:40 mark) First gap = 1 units, Second gap = "mystery number" of units (x). (0:05 mark) By comparing the two calipers, we can see that "growing" First gap to the size of Second gap (multiplying 1 by x) causes Second gap to grow to the size of the Third gap (multiplying x by x). (Back to 1:45 mark) Thus, we can label Third gap = "x^2". The other powers follow the same logic; expanding the calipers by a set ratio gives us the values of all the gap relative to gaps we already know.

[@OpinionAxe]

The real MVP is always in the comments.. thanks!!

[@hankroest6836]

My new favourite Numberphile video! Wow. The star even has the same name as one of my favourite actors! ;-)

[@NetAndyCz]

Why we do not want the complex solutions? Is it because they are hard to visualize or they do not work in the original equations?

[@Taintain101]

Im kind of surprised you can't expand to however many pins you want, it would be cool to see a proof of that, or more plastic number stuff in general. Really interesting video :)

[@redbeam_]

What even are these calipers? Are they used for something? I've never seen them before...

[@jroemling]

Adam Savage is a Patreon? Nicely played!

[@papawheely3627]

All this talk about the ratio creating powers of itself, and no one thinks to call it "the most powerful ratio".