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

In summary, the conversation is about different forms of the golden ratio that are not rewrites of the commonly known expression. The person has searched on Google and checked Wikipedia and Wolfram, but has not found any new forms. However, they have found a general form involving trigonometric functions. The conversation ends with a request for others to share any additional forms they may know.
  • #1
mesa
Gold Member
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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
 
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  • #3
micromass said:
http://en.wikipedia.org/wiki/Golden_ratio#Alternative_forms

A good link, but I had already seen it. The trigonometric versions are just substitutions of the same expression, ø = 1/2(5^(1/2)+1).

I also checked out Wolfram and I noticed this,

ø=2^(-1/v)(5^(1/2)Fv+(5(Fv^2)-4cos(v∏))^(1/v)

which is a wonderful general form!

Anyone else have anything to add?
 

What is the golden ratio?

The golden ratio is a mathematical ratio that has been observed and studied for centuries. It is approximately equal to 1.618 and is often represented by the Greek letter phi (φ).

What are some well-known examples of the golden ratio in nature?

Some commonly cited examples of the golden ratio in nature include the spiral patterns found in seashells, the branching patterns of trees, and the proportions of the human body.

Can the golden ratio be found in art and architecture?

Yes, the golden ratio has been used in art and architecture for centuries. It is believed to create aesthetically pleasing and harmonious designs. Examples include the Parthenon in Greece and the Mona Lisa painting by Leonardo da Vinci.

What is an exact form of the golden ratio?

An exact form of the golden ratio is a number that is equal to the golden ratio, such as 1.618. Other exact forms include the continued fraction representation of [1; 1, 1, 1, 1, 1, ...] and the algebraic representation of (1 + √5)/2.

Can the golden ratio be found in music?

Yes, the golden ratio has been used in music composition and theory. Some examples include the length of musical phrases and the positioning of musical notes on a staff. However, its use in music is debated and not universally accepted.

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