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,
a
+
b
a
=
a
b
=
def
φ
{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi }
where the Greek letter phi (
φ
{\displaystyle \varphi }
or
ϕ
{\displaystyle \phi }
) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation
x
2
−
x
−
1
=
0
{\displaystyle x^{2}-x-1=0}
, with a value of:
φ
=
1
+
5
2
=
1.618033
…
{\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.
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 =...
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...
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...
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...
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...
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!
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?
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...
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...
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...
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...
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
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...
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...
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...
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.
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.
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?
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...
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...
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...
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.
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...
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...
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...
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.
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)...
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...
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...
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...
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...
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.
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