InWhat Situations Do We Enounter the Golden Ratio?

[musicgold]

Homework Statement

Not a homework problem.
What kind of situations/problems are likely to throw up the golden number?

Relevant Equations

I know what the golden ratio is and am generally familiar with Euclid's work on the number.
As we know, when two quantities, both a function of the same variable, are multiplied, we are bound to get the quadratic function. Similarly, when is one bound to see a golden ratio situation?

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 read more on such topics.

 

[Delta2]

I am afraid I can't answer directly your questions, probably a more knowledgeable member than me can, but did you read the entry of wikipedia?
https://en.wikipedia.org/wiki/Golden_ratio
It has a ton of references too there.

I know wikipedia articles are usually of average quality but this seems good.

 

[neilparker62]

Homework Statement:: Not a homework problem.
What kind of situations/problems are likely to throw up the golden number?
Relevant Equations:: I know what the golden ratio is and am generally familiar with Euclid's work on the number.
As we know, when two quantities, both a function of the same variable, are multiplied, we are bound to get the quadratic function. Similarly, when is one bound to see a golden ratio situation?

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 read more on such topics.

Pivotal in the development of early trig tables - ratios of pentagon related angles such as $18^{\circ}, 36^{\circ}, 54^{\circ}$ and $72^{\circ}$. Used in conjunction with Ptolemy's theorem.

 

[fresh_42]

The golden ratio occurs often in nature, and in architecture. And the internet is full of references under the search key "golden ratio". Are you looking for something particular?

 

[Mark44]

As we know, when two quantities, both a function of the same variable, are multiplied, we are bound to get the quadratic function.

No, we don't know this. It's true only if the quantities are linear functions in the shared variable, not for arbitrary functions.

I am interested in knowing the kind of problems the ancient Egyptians and Greeks were grappling with when they encountered the golden number.

At the risk of repeating what can be found in the Wikipedia article, the Greeks were trying to find the most aesthetically pleasing dimensions for a rectangle, such as the shape of the front of temples. They decided that the dimensions should be the ratio of the width to the height should be equal to the ratio of the height to the sum of width and height. I.e., W : L :: L : W + L. If we set W = 1, this leads to the quadratic equation $L^2 - L - 1 = 0$. The larger root is called the golden ratio and has been given the name $\phi$. Its value is approximately 1.618.

Successive terms in the Fibonacci Sequence approach $\phi$ in the limit.

 

[musicgold]

No, we don't know this. It's true only if the quantities are linear functions in the shared variable, not for arbitrary functions.
At the risk of repeating what can be found in the Wikipedia article, the Greeks were trying to find the most aesthetically pleasing dimensions for a rectangle, such as the shape of the front of temples. They decided that the dimensions should be the ratio of the width to the height should be equal to the ratio of the height to the sum of width and height. I.e., W : L :: L : W + L. If we set W = 1, this leads to the quadratic equation $L^2 - L - 1 = 0$. The larger root is called the golden ratio and has been given the name $\phi$. Its value is approximately 1.618.

Successive terms in the Fibonacci Sequence approach $\phi$ in the limit.

Thanks.

The most articles on the golden ratio on the Internet are about what the golden ratio is, and why it is aesthetically pleasing, and how the ratio of the Fibonacci numbers approach the golden ratio.

I, however, feel that these properties of the golden ratio were probably discovered afterwards and that the ancients probably encountered this "weird" quantity while working on something else. For example, how the Greeks encountered the irrational sqrt(2) while trying to quantify the hypotenuse of a 1x1 square, or how they struggled when they tried to double the volume of a cube.

 

[fresh_42]

The major point is that in the world of the ancient Greeks, the length of a line corresponded to a number, and the relation between two numbers had been a point that separated such a line (of final length) into two sections. This was what they primarily dealt with. To understand the golden ratio in this regard, have a look at the Wikipedia article https://en.wikipedia.org/wiki/Golden_ratio but do not read it. Look only at the images.

 

[Mark Harder]

One of the roots of $x^2 -x -1$ is $\frac {1 + \sqrt 5} 2$ and that = $\Phi$ .

 

[Mark44]

One of the roots of $x^2 -x -1$ is $\frac {1 + \sqrt 5} 2$ and that = $\Phi$ .

Usually written as lower-case: $\phi$.

 

[WWGD]

I dont know if Im mixing things up here, but Im confused that the Greeks considered Irrational numbers somehoe unnatural, re the whole issue with $\sqrt 2$ ,somehow include $\sqrt 5$ in a formula for beauty, or even for " Elegance ".

 

[fresh_42]

I dont know if Im mixing things up here, but Im confused that the Greeks considered Irrational numbers somehoe unnatural, re the whole issue with $\sqrt 2$ ,somehow include $\sqrt 5$ in a formula for beauty, or even for " Elegance ".

Natural or rational numbers have always been lengths and ratios of lengths to the ancient Greeks. It is all about ratios. The golden cut is defined as a ratio: $\dfrac{a}{b}=\dfrac{a+b}{a}.$

On the other hand, $\sqrt{2}$ is $\dfrac{a}{b+b}=\dfrac{b}{a}.$