What is Golden ratio: Definition and 38 Discussions

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,








{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi }
where the Greek letter phi (


{\displaystyle \varphi }


{\displaystyle \phi }
) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation





{\displaystyle x^{2}-x-1=0}
, with a value of:






{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.618033\ldots }
The golden ratio is also called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio, medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.
Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.

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  1. M

    B Golden Ratio in Collatz-like sequences

    Consider the following: We start with a positive integer: x If x is even, do x/2 If is odd, do Floor function( x * y) with y being some decimal number between 1 and 2 And repeat until a loop is reached. If 1 is reached, the next number will be 1 as well. So we reach a loop too. An example: x =...
  2. MevsEinstein

    B Popularizing a property for n-bonacci numbers without publishing it?

    Hi PF! Everyone knows that: $${\varphi }^2 - \varphi - 1 = 0$$ But guess what? $${\varphi}^3-2{\varphi}^2+1=0$$ Generalizing this for all n-bonacci numbers: $$x^{n+1}+1 = 2x^n$$ where ##x## is the n-bonacci number and ##n## is the degree of the polynomial that the n-bonacci number is a root of...
  3. M

    InWhat Situations Do We Enounter the Golden Ratio?

    In fact, the whole golden rectangle and ratio discussion seems strange to me. I am interested in knowing the kind of problems the ancient Egyptians and Greeks were grappling with when they encountered the golden number. It would be great if you could point me to a book or website where I can...
  4. Prinzmio

    Lowering the Golden Ratio: The Impact on Golden Section Search Efficiency

    Currently my task is to count number of iterations of golden section search verus interval bisection search of a function y = x^2. Golden section search took about twice the number of iterations than interval bisection search. If I lowered the golden ratio from 0.618 to 0.5562 , the number...
  5. kshitij

    Quadratic equation and its roots

    On simplifying the given equation we get, x^2-x-1=0 and using the quadratic formula we get x=(1+√5)/2 and x=(1-√5)/2 Now, as the formula suggests, there are two possible values for x which satisfies the given equation. But now, if we follow a process in any general calculator by entering...
  6. M

    I Exploring the Golden Ratio in Hexadecimal: How to Calculate the First 50 Digits

    I am not sure if this is the right section for this, but i didn't find it anywhere and I really need it. Can someone point me towards a place/tell me a program where I can get/compute the first (at least) 50 digits in the hexadecimal representation of the golden ratio? Thank you!
  7. M

    A Golden ratio fractal phase conjugation

    So, I recently heard about the Golden ratio fractal phase conjugation and I was wondering if it could possibly explain the solar system formation. Meaning, if you can imagine a torus energy dynamic shape (with the sun in the middle) maybe it could be explained by this theory?
  8. nomadreid

    I Physically relevant: fractals, phi?

    There are two subjects which pop up a lot as having physical examples (or, more precisely, where their approximations have), but many (not all) of them seem rather indirect or forced. For example: [1] phi (the Golden ratio) or 1/phi: (a) trivia: sunflowers and pineapples giving the first few...
  9. DaTario

    I 360 degrees and the Golden Ratio

    Hi All, I have just found in the internet an identity showing that the Golden Ratio can be expressed as a function of the cosine of the angle of 36 degrees. It seemed to me as an important fact related to this specific angle. Had this fact, historically, any relevance to the choice of the 360...
  10. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  11. TheBiologist

    Understand the Golden Ratio - Get Help Here

    I've heard of it, but unfortunately, I don't really understand in depth what it actually is. Could someone please help me with this? Thanks.
  12. H

    Is the golden ratio really the solution for Tesla coil voltage issues?

    Hello all, I am in the process of designing a tesla coil for a research paper, and, just when I thought I had seen all the crack science there could possibly be, I came across a claim concerning the golden ratio. (The natural progression of events, I guess...) There is a real problem when...
  13. W

    MHB Help Me Solve the Golden Ratio Question

    Question is attached. I don't know how to find the golden ratio I really did try to solve this one, but i couldnt please please help me out
  14. mesa

    What are some examples of exact forms of the golden ratio?

    Hello all, I am looking for exact forms (as real number expressions) of the golden ration that are not rewrites of the one we all know and love, i.e. g.r. = 1/2(5^(1/2)+1) Searches in Google have yielded nothing so far :P
  15. E

    Golden Ratio Dragon Fractal (figured it out)

    Back in november I asked the forum about this fractal: http://en.wikipedia.org/wiki/File:Phi_glito.png At the time I couldn't figure out how to make it. Since then I've figured it out. I used MS Excel. I'm not completely satisfied though. There are some gaps between the major...
  16. E

    Is there a connection between Rule 30 and the Golden Ratio?

    I was messing around with cellular automata and wanted to test a theory i had. If you look close at the elementary cellular automata Rule 30 you'll notice that the right side is chaotic, while the left side has chaotic properties but it does have order and repetition. There is an imaginary...
  17. E

    Dragon Curve Fractal Using Golden Ratio

    I've been fooling around in MS Excel trying to reconstruct this fractal: I haven't had any issues here making it. I totally understand the algorithm for generating the left turn/right turn ordering. What I really want to know is how this version is generated: Original image...
  18. B

    Need help w/ Fibonnaci and Golden Ratio proof

    Let F_n and F_n+1 be successive Fibonnaci numbers. Then |(F_n+1)/(F_n) - Phi | < 1/(2(F_n)^2) Where Phi is the Golden ratio.
  19. D

    Trignometric and hyperbolic equalities: Why the golden ratio?

    1. \sin \theta = \cos \theta \theta=\frac{\pi}{4} 2. \sin \theta = \tan \theta \theta = 0 3. \cos \theta = \tan \theta \theta =\arcsin (\varphi -1) 4. \sin \theta = \csc \theta \theta = \frac{\pi}{2} 5. \sin \theta =\sec \theta \theta does not exist. 6. \sin \theta =\cot...
  20. K

    How do you construct the pattern of a dragonfly wing using the golden ratio?

    You can see it here starting at 2:30 https://www.youtube.com/watch?v=http://www.youtube.com/watch?v=kkGeOWYOFoA My question is, how do you determine the position of the points used to draw the triangles and circles.
  21. M

    Is There a Connection Between Quantum Physics and the Golden Ratio?

    Hello, this is my first post so please be patient. I studied mathematics and have actuary qualifications. I was always obsess with the golden ratio. I was wondering if there was a relation between quantum physics and the golden ratio. A hunch told me so.
  22. E

    What's the perfect math tattoo for me?

    I've been thinking of getting a math-related tattoo, and I decided to go ahead with it, but I'm not sure what it should be. Maybe a small Euler spiral on my wrist? My mother thinks I should get the rational version of the golden ratio, but I think that'd just be lame. Thoughts please?
  23. Helios

    Solving for x: Uncovering the Golden Ratio

    If tan x = cos x, then what is x ? The answer includes the golden ratio !
  24. G

    Medical Exploring Memory Span, Quantum Thought & the Golden Ratio

    Memory span as the "quantum" of thought plus golden ratio the basis of everything? I found this paper linked to in wikipedia's article on information theory. After a little bit of background checking, it seems the authors (or at least one of them) is known as being controversial for their...
  25. M

    Phi (1.6) Found at Quantum Level: Major Breakthrough?

    So I've read that phi (1.6) has been found at the quantum level, and from what I've been told this is a major breakthrough. But I fail to see the relevance. I was wondering if anyone felt that this is a major breakthrough, and could explain this to me? edit: posting link to article...
  26. M

    Understanding the Golden Ratio: Exploring \phi and \varphi in Mathematics

    Im working on a part off my course and it covers this, but its not clear. \phi= half (1+\sqrt{5}) \varphi=half (1-\sqrt{5})The question asks \phi-\varphi =\sqrt{5} It is written in my book, the answer but it does not explain how the maths cancels and manipilates. Could you show me a way that...
  27. A

    Fibonacci series and golden ratio

    I was wondering if someone could explain the fibonacci series and golden ratio to me, I'm very curious, but I don't have that much experience in math as a high school trig student.
  28. G

    Golden ratio base useful? Easy logarithm in phinary system

    I was wondering if the Golden ratio base (phinary system) has any use somewhere and if arithmetics with it is easy? I programmed a surprisingly simple algorithm to calculate the logarithm yielding digits in base phi using nothing more than 2 multiplications/divisions per result digit. Can it...
  29. A

    Phi (the golden ratio) in prime quadruplets

    Phi exists at the center of prime quadruplets, along with its square root, and cube root! http://www.code144.com/zphithrice.png The 'pos' numbers come from the position of the prime numbers in the sequence itself, i.e. 193 is the 44th prime number, and 197 is the 45th prime number...
  30. G

    Accuracy, Fibonacci + Golden Ratio

    I have been curious about this for a while... I'm interested to know if there is any easy way to tell the accuracy of the (n+1)th on the nth term of the Fibonacci series in relation to the golden ratio. I know that as n tends to infinity the ratio tends to the Golden Ratio "Phi" - but is...
  31. madmike159

    Explain the Golden Ratio: (a+b)/a=a/b

    Can someone please explain the golden ratio to me. I looked on wikipedia (http://en.wikipedia.org/wiki/Golden_ratio) for an explanation but I couldn't make sense of it. How does (a + b) / a = a/b. What does a+b is to segment a, as a is to the shorter segment b.
  32. G

    Blueshift + Classical Physics = Golden Ratio

    I was doing calculations to see how far classical physics would take us in terms of the speed of an object never exceeding the speed of light in a reference frame. Here was the scenario I set up: http://la.gg/upl/light.jpg So, if we want to find the time it takes the light to get from (A)...
  33. R

    Construction of a cyclic sequence re the Golden Ratio

    The fractal sequence http://www.research.att.com/~njas/sequences/A054065 is of interest because it provides permutations of the numbers 1-n such that the decimal part of k*tau (k = {1,2,3,...n} is ordered from the lowest possible value to the highest. For instance if n = 3 the permutation...
  34. P

    Proving the Golden Ratio for a W-Shaped Quartic Function

    I formed the following statement: A "W"-shaped quartic function f(x) has two points of inflection B and C. A line through the points B, C passes through f(x) again at A and D. The ratio AB:BC:CD simplifies to 1 : \phi : 1. So, AB = CD and \phi = 1.61803399... , also known as the golden...
  35. E

    Infinite 45 degree golden ratio thingy series related

    I've been trying to figure this problem out for hours on end and i can't even go on the tinernet to find the answer because its hard to search for. The problem is: You start off at hte coordinate (4,0). you move up 45 degrees and the distance traveled is now divided by the square root of 2...
  36. F

    Fibonacci Phi - The Golden Ratio

    Anyone else fascinated with the Golden Ratio (Phi)? It seems that there is an underlying principle with everything that is in this world that has some sort of aspect related to Phi. From artwork, proportions of the human body, to the growth rate of biological cells. Everything seems to have...
  37. K

    Is There a Relationship Between the Golden Ratio and Physics?

    Are there any connect between the golden radio( or any well know constant like the fibonacci number) and physics? I ask this question because there are a lot these occurence in math.
  38. T

    Relationship between triangle and golden ratio

    I am having trouble finding the relationship between a 72-72-36 triangle and the golden ratio. Could someone point me in the right direction or explain it? Thanks