what is the plastic ratio?
HTML-код
- Опубликовано: 30 июл 2024
- what is the plastic ratio? I define the plastic number and explain its properties, first as a cubic equation and a Fibonacci sequence to count bunnies, then as a fractal cube root with self similarities, and finally a cool formula with hyperbolic cosine function substitution. This is a cool journey through algebra, polynomials, inverse functions, and calculus.
Proof that I did it before mathologer: • Plastic Number
0:00 Introduction
0:36 Fibonacci
1:45 Fractal
3:10 Hyperbolic trig
Calculating hyperbolic trig functions: • Can you simplify this ...
YT channel: / drpeyam
TikTok channel: / drpeyam
Instagram: / peyamstagram
Twitter: / drpeyam
Teespring merch: teespring.com/stores/dr-peyam
I love cosh³(x) and sech³(x) trig identities :p they are very important/handy in self-trapping problems, like describing polarons (self-trapped electrons in a metal). Didn't know the plastic ratio has them too, pretty cool indeed :D
in base phi, it's possible to express all integers in a finite form, which is really rare for an irrational base:
1->1, 2->1.11=10.01, 3->11.01=100.01, 4->101.01.
An interesting property, of course, being that no number expressed in base phi ever needs to have adjacent ones, due to the relation: 100 = 11.
So what about base rho?
sqrt = lambda i: i**0.5
cuberoot = lambda i: i**(1/3)
rho = cuberoot(1/2 - sqrt(69)/18) + cuberoot(1/2 + sqrt(69)/18)
rho**3 == rho+1
so in base rho, 1000 = 11, which means...
1->1, 2->100.00001, 3->101.00001, 4->10000.1000000101, 5->10100.0000000101
1 takes 1, 2 takes 2, 3 takes 3, 4 takes 4, 5 takes 4, 6 takes 5, and i think 7 cannot be written in non-adjacent form in finitely many digits.
I have no idea why you would think that 7 couldn't be written in non-adjacent form in finitely many digits (the substitutions 0011=1000 and 00200000=10000001 will inevitably create a non-adjacent form, though they grow in length much more quickly than in base phi)
In any case, I worked it out by hand and got 1000000.1010100101 for 7, which I confirmed using a calculator.
I love how if cosh were to be linear for scalars you could take that 3 out the inner arccosh, cancel it with the 1/3, then cancel the cosh and arccosh between themselves, and finally the fractions to just get 1! So tempting!
It's all good, Dr. Peyam. I checked and verified that Mathologer is _not_ mentioned in the Plastic Number page on both Wikipedia and Wolfram MathWorld.
Unfortunately, they don't mention you either 😢. So if you confirm that you came up with it before Gérard Cordonnier in 1924, I'll make the necessary edit to Wikipedia and submit a correction request to Wolfram. 👍
As an engineer I had an applied problem that caused
me to end up generating fantastic hyperbolic formulas.
But you made me laugh as I had the same exact thought
when you wrote down and said who in the world does
arc-hyperbolics. LOL (Apologies to those who use arc-hyp's.)
Arc hyperbolic come up in physics multiple times....the solution to quadratic drag forces, and the rapidity of special relativity are the two cases that immediately spring to mind.
Enjoyed this much. Thanks. Hadn't played with that one before. Cheers!
(funny how sometimes something seeming complex can yield a very simple answer, and something seemingly simple can be made to yield that monster!)
If you use the cubic formula you get rho = cbrt((9+sqrt(69))/18) + cbrt((9-sqrt(69))/18) 👍
Nice!
"If you use the cubic formula..."
No thanks.
@@DeJay7 haha it wasn’t too bad because the coefficient b(=0) appeared quite a lot so there wasn’t much left.
@@dethmaiden1991 Since it has no quadratic term, the cubic polynomial p³-p-1 is referred to as a "depressed" cubic, and this makes solving it soooo much easier (also that it's monic i.e. has leading coefficient 1 helps).
I was wondering if that number was even real, but with the cubic formula we see that it actually is.
The decimal approximation is 4.15895115
Plastic ratio,love it!! I wonder if they are more nice ratios like this.💯💯
One could replace √5 in Phi = (1+√5)/2 by √n :
Name = (1+√n)/2
And replace Phi² = Phi+1 by:
Name² = Name+(fn)
If for example n=6 then (fn)=1.25
n=7 => 1.5 etc...
(fn) = (n-1)×0.25
Awesome 👍👍
Note cosh and arc cosh are going from way of solving cubic equation.
Best way to calculate such kind of ratio numerically are Newton's method or Chebyshev's method.
Polynomials are most easiest for them.
True
I discovered this constant myself around 9th grade, from a spiral made of equilateral triangles, analogous to the golden ratio in the Fibonacci spiral. I was quite proud at the time of the new interesting number found.
Very cool!
earned a sub
Mornin Dr Peyam, long time without looking at your 's tutoriales,,now i'm again,thanks You for share You knowledgement,and sorry because of My por English,likely maths is universal idioma.
Regards,Doctor!
Can you turn it into a Taylor expansion?
i remember your original video about this and even commenting under it
thank you mr , can you talk about hadamard-stieltjes fractional integral equation please
There’s a video on Stieltjes integration already
@@drpeyam I will look for it thank you very much
Why did you assume That Rho is equal to the cosh(theta) term? Any explicit reason?
I learnt about the plastic number from Ian Stewart's book "Math Hysteria". I highly recommend that book.
I love Ian Stewart!!!!
As a polymer physicist/chemical engineer … I got very excited that there might be some applications. Still nifty though
I’m assuming there’s no way to get the cosh and arccosh to cancel?
Nope
The video hit 50k views-- Time for Dr Peyam to sing the Pokémon Song! ruclips.net/video/8cr2a18hxu8/видео.html
Strangely when you use the linear algebra matrix for the Plastic Fibonacci analogue the matrix Is the same with 1 in all entries except bottom right zero giving the same phi as the eigenvalue . However the recurrence is now shifted back to previous second and third terms which l suppose evolves into the Cosh Plastic Ratio .
Hey Doc you've found a way to make Plastic Bunnies 🐇🐇🐇🐇....your favourites haha
Where's my requested Theta functions involved in establishing the Zeta Functional Equation ?
How or why did you define rho to be (2/3^1/2)cosh theta..??❤ Bill
As I mentioned on the Nummerology Site, Ian Stewart has demonstrated this Ideas in one of his Books many Years ago.
I do not know who triggered whom, but it is old knowledge.
nice
What is cosh ? Ik like sin , cos , tan , sec , cosec and cot ... is it another name for one of this ?
No it’s one half of exp(x) + exp(-x)
@@drpeyam thank you Sir :)
Sweet memories with your Tshirt.❤️❤️
You could use Cardan's method
rho=s^(1/3)+t^(1/3)
rho^3 = s + t + 3 rho (st)^(1/3)
= 1 + rho
=> s+t =1
st = 1/27
(s - t)^2 = 1 - 4st - 1 - 4/27 = 23/27
s - t = sqrt(23/27)
s =(1 + sqrt((23/27)))/2;
t =(1 - sqrt((23/27)))/2;
rho = ((1 + sqrt(23/27))/2)^(1/3) + ((1 - sqrt(23/27)/2)^(1/3)
= 1.3247 approx
How can the root of a polynomial equation with integer coefficients be a transcendental number?
It’s not transcendental
ofcourse it's not transendental! he literally just derieved it from a an equation! That's the very definition of transendental. Do you even know the difference between irrational and transendental?
@@aryadebchatterjee5028 Bad English: DERIVED! TRANSCENDENTAL!
@@aryadebchatterjee5028 chill out, man
The cosh and arcosh undo each other enough to make it algebraic. Another example is sin(arcsin(1/2)/3), which is algebraic, but not constructible.
♥great
Is there a formula for cosh(AB)?
Not that I know of
There is one for cosh(2t) but not sure it helps clean up. “When in life do you use arccosh(x)?” 🤣 That was hilarious.
Punched that on my calculator, got the value for rho, took the third power and the result increased by 1. It worked!
Nice ,tell us about what's written on your shirt !
There's no such thing as ARCcosh. It's ARcosh, that is, the Hyperbolic ARea COmplementary Sine function. Since it's area cosh, it's not arc cosh, so we get arcosh as the abbreviation. The full name of the regular arccos function is, in case you wondered, [circular] ARC COmplementary Sine function. As for why it's area vs arc, just think about what's happening geometrically. The hyperbolic functions represent the area under a curve, whereas the circular functions represent an arclength.
Interesting
Why not express it with radicals
This value can be expressed by real radicals
because it is not irreducible case
Approximately what rho equal to?
Graphing the equation you get the intersection at about x= 1.325
Wolfram Alpha says it's about 1.3247179572447460259609088544780973407344040569017333645340150503... Hope that's enough digits ;p
~ 4/3 since the exact value is a + 1/3a, where a = [3]√((1/(18))√3√(23)+(1/2)) ~ 1 ;)
I ran it through the cubic formula and got cbrt((9+sqrt(69))/18) + cbrt((9-sqrt(69))/18) which matches Joan’s answer.
Gary Austin's formula can give you a precise value to as many digits as your calculating device will allow. As an alternative, if you have a calculator with an "Ans" key and a "³√" key, you can use an iterative formula:
Press "1" then press "=".
Type in "³√(1+Ans)", then press "=" repeatedly until the result stays constant somewhere near 1.324717957244746. The rate of convergence is a bit slow, but it works well.
You can just about do it with the Windows calculator, but it's a much more elaborate and drawn-out process.
"I'm just playing games
I know that's plastic love"
The inverse function of cosh is called "area cosh", not "arc cosh".
❤❤❤❤❤
The plastic ratio would be way cooler if it was used to calculate the perfect proportions for breast augmentation.
Boss Vibes thanks
Good job! Hilarious.
Interesting, using the hyperbolic trigonometry
isn't familiar to me for solving 3rd degree equation.
Using classic trigonometry is more well known
in the case the discriminant is 0
sqrt(D)=1/2sqrt(23/27)
D>0 then E admits only one real solution such as
x=u+v with
u=(-q'+sqrt(D))^(1/3)
and
v=(-q'-sqrt(D))^(1/3)
then
u=[1/2(1+sqrt(23/27))]^(1/3) #0,986991206
and
v=[1/2(1-sqrt(23/27))]^(1/3) #0,337726751
Then x=u+v # 1,324717957 as the plastic ratio
Which leads to the same approx value when computing video formula
x=2/sqrt(3)cosh[1/3arccosh(3sqrt(3)/2)]
Why you call it a plastic ratio?
Because of the golden ratio
Anyone like to know about the architectural origin?
What is intersting about the answer we got?
When cubed, it gets exactly one bigger.
phương trình bậc hai.
That shirt. 😂😂
Isn't this just a specific instance of a lagged Fibonacci sequence?
That’s what I mentioned in the video
Thank you for pronouncing φ correctly 🤪
It's the ratio of growth if the n-1 bunnies have to go through puberty first
That’s what it is!
Maybe I'm First here
Yahoooo 🥳🥳🥳
Nice video, but it does not explain the question in the title of the video.
Yes it does?
The wooden ratio
But what is its numerical value (approximately)? 🤔 I know pure maths lovers don't care about that 🤣
So... What is the plastic ratio? You didn't answer this. We know, what is golden ratio, what meaning it has for the nature. But what the ... plastic ratio, YOU DIDN'T ANSWER. Just clickbate, I'm very disappointed.
It’s the plastic ratio
"Fee" 🤣🤣🤣🤣🤣
I see that someone's been learning modern Greek pronunciations of a few selected alphabets. 🙄
Why not be consistent and pronounce "π" the modern Greek way as well 😜? At least then, American high school students will have a lot more fun learning math! The icons for π-day will also get a much-needed replacement.
The problem with that is that it sounds like the English name of 'p', except for aspiration, which is not phonemic in English.
Phee for phi sounds horrible in England. Same with bayta, thayta. I call this "modern" thing wikipedantry. You don't have to spell and say everything the same way as the source language. (To be fair the US pronunciations are probably inherited from French or German, not modernised in that sense).
Actually when I had students they pronounced both phi and theta as "thi" (long i). Compromising I guess :-))