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

• madmike159
In summary, the golden ratio is a mathematical concept defined by the rule that (a+b)/a = a/b, where a and b represent segments. It can also be found by solving problems involving numbers with a difference of 1, using the Fibonacci sequence, or by constructing a golden rectangle. The golden ratio is considered to be the most irrational number and is said to have the most elegant proportion of any rectangle.
madmike159
Gold Member
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.

madmike159, you might become more comfortable if you focus on how the ratio value relates to the fibonnaci sequence; and then examine the other parts of the article in Wikipedia. Also study the relevant sections from a high school Geometry book.

There's nothing really to explain.

The golden ratio a/b is defined by the rule that a/b = (a+b)/a. If we set b=1, then a=(a+1)/a, and we see that it is the larger of the two roots of x^2-x-1.

Sure, the golden ratio is such that (a+b)/a = a/b. Now let's work this equation. First let's multiply both sides by a to get (a+b)=a^2/b now let's multiply both sides by b to get b(a+b) = ab + b^2 = a^2 now we rearrange to get b^2 + ab - a^2 = 0 or a^2 - ab - b^2 and this is simply a quadratic formula so we want to find out for what values of a this is valid, thus using the quadratic formula we get a = (b +/- sqrt(b^2+4b^2))/2 which equals b*(1 +/- sqrt(5))/2 but since 1 - sqrt(5) is negative (and there's no such thing as a negative length) the only solution of interest is a = b*(1 + sqrt(5))/2 and by a quick rearrange we get a/b = (1+sqrt(5))/2 the pseudo-mystical golden ratio

Remember that a and b both represents segments. You can think of a+b as the length of a plus the length of b. If you picture a and b connected at an endpoint than a+b becomes one large line-segment that can be split into two line-segments a and b. Then a and b are in a sense sub line-segments.

Continuing with that idea, then you can re-write this: " a+b is to segment a, as a is to the shorter segment b."
As
The total lineament is to the longer sub line-segment, as the longer sub line-segment is to the the shorter sub line-segment.

I hope that made sense.

Thanks I understand now. I was mostly confused how they got from (a+b)/a to a/b.

I used to think he golden ratio was a lame constant whose entire importance is related to the fact that it shows up in some DEs. However, it actually is a pretty interesting number, and here's why - the golden ratio, phi, can be defined as the number whose reciprocal is equal to 1 minus itself.

phi-1 = 1/phi.

Now, that's all fine and good, but something interesting happens when you take the continued fraction expansion:
phi = 1+1/phi = 1+1/(1+1/phi) = 1+1/(1+1/(1+1/phi)) = 1+1/(1+1/(1+1/(1+1/(...))))

It is the only number whose continued fraction expression is 1 1 1 1 1 1 1 1...

This can be used to show that phi's rational approximation converges as slowly as possible, meaning that in some sense that phi is "the most irrational number". Now that is interesting!
http://www.ams.org/featurecolumn/archive/irrational1.html

The golden ratio can be found by solving the following problem :

Find two numbers that have a difference of 1, and when multiplied together equal 1.

The numbers are 1.61803... and .61803... ; the golden ratio being former.

Another way to find the golden ratio is to use the Fibinachi sequance, where every term is the sum of the previous two terms, starting with 1,1

1,1,2,3,5,8,13,21,34,55...

The golden ration can be approximated by the ratio of any two succesive numbers in this sequance. The approximation gets better the larger numbers you use.

ie. 55/34 = 1.617...

The "golden rectangle" (it is said that the side view of the Parthenon in Athens is a golden rectangle and that DaVinci's painting "The Last Supper" is in the proportions of a golden rectangle. More generally it is claimed that the golden rectangle has the "most elegant" proportion of any rectangle. That is, of course, not a mathematical claim.) is a rectangle, with width w and height h, such that, if you mark off distance h from one corner of the width and draw a perpendicular to make a new rectangle; that is, construct a new rectangle having width w-h and height w, the ratio of "height to width" is still the same: you have constructed a new golden rectangle.

The "height to width" ratio of the first rectangle is h/w, the "height to width" ratio of the second is w/(w-h). If those are the same h/w= w/(w-h). Taking a= w-h and b= h, then a+ b= w-h+ h= w so h/w= (a+b)/b and w/(x-h)= b/a: the proportion h/w= w/(w-h) is (a+b)/b= b/a.

From h/w= w/(w-h) we can multiply both sides by w(w-h) and get h(w-h)= w2 or hw- h2= w2. Dividing both sides by h2, (w/h)- 1= (w/h)2. (w/h)2- (w/h)+ 1= 0. Using the quadratic formula to solve that equation gives phi as the positive solution for the ratio w/h.

HallsofIvy said:
The "golden rectangle" (it is said that the side view of the Parthenon in Athens is a golden rectangle and that DaVinci's painting "The Last Supper" is in the proportions of a golden rectangle. More generally it is claimed that the golden rectangle has the "most elegant" proportion of any rectangle. That is, of course, not a mathematical claim.) is a rectangle, with width w and height h, such that, if you mark off distance h from one corner of the width and draw a perpendicular to make a new rectangle; that is, construct a new rectangle having width w-h and height w, the ratio of "height to width" is still the same: you have constructed a new golden rectangle.

The "height to width" ratio of the first rectangle is h/w, the "height to width" ratio of the second is w/(w-h). If those are the same h/w= w/(w-h). Taking a= w-h and b= h, then a+ b= w-h+ h= w so h/w= (a+b)/b and w/(x-h)= b/a: the proportion h/w= w/(w-h) is (a+b)/b= b/a.

From h/w= w/(w-h) we can multiply both sides by w(w-h) and get h(w-h)= w2 or hw- h2= w2. Dividing both sides by h2, (w/h)- 1= (w/h)2. (w/h)2- (w/h)+ 1= 0. Using the quadratic formula to solve that equation gives phi as the positive solution for the ratio w/h.

Wow that is confusing. I'm going to have another look in the moarning =D. Thanks for the help every one. I think I know enough for what I need to do, but any more places where the GR appears would be nice to know (wiki article said something about the golden ratio in trees and humans =S).

madmike159 said:
Wow that is confusing.
It's actually very simple. It's a rectangle which you try to make square by cutting out the excess material. Try to picture:
• a square of dimensions a x a;
• a rectangle which is (a+b) x a (so that the square fits in);
• the remainder of the rectangle, after you cut out the square, which is a smaller rectangle of dimensions a x b.
If the big rectangle is proportional to the smaller one, the ratio of their sides, (a+b) / a = a / b gives you the golden ratio.

Try this:
http://golden-rectangle.lcpdesign.com/construct.htm

Last edited by a moderator:
Yea it is really easy. Best not to do maths late at night though. Thanks for the help everyone.

## 1. What is the Golden Ratio?

The Golden Ratio, also known as the Divine Proportion or Phi, is a mathematical ratio that is approximately 1.618. It has been studied and admired by mathematicians, artists, and scientists for its aesthetic and harmonious properties.

## 2. How is the Golden Ratio expressed?

The Golden Ratio is often expressed as (a+b)/a = a/b or as the decimal value 1.6180339... It can also be written as the Greek letter phi (φ).

## 3. Where can the Golden Ratio be found in nature?

The Golden Ratio can be observed in many natural phenomena, such as the spirals in seashells, the branching patterns of trees, and the proportions of the human body. It is also seen in the growth patterns of plants and the arrangement of leaves on a stem.

## 4. How is the Golden Ratio used in art and architecture?

The Golden Ratio has been used by artists and architects since ancient times to create aesthetically pleasing compositions. It can be seen in the proportions of famous works of art, such as Leonardo da Vinci's "Mona Lisa" and in the design of famous buildings, such as the Parthenon in Greece.

## 5. Is the Golden Ratio present in music?

Yes, the Golden Ratio has been used in music composition and is believed by some to create a more pleasing and harmonious sound. It can be found in the proportions of musical scales, the length of musical notes, and the structure of musical compositions.

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