# 360 degrees and the Golden Ratio

• I
• DaTario
In summary, the Golden Ratio can be expressed as a function of the cosine of the angle of 36 degrees, which may have some historical relevance to the choice of the 360 degrees standard. However, the sexagesimal system, which is much older than trigonometry, is the likely basis for the number of degrees in a circle. Natural proportions, possibly anatomic, may have played a role in this connection. The role of the number of days in a year adds further confusion to this discussion.
DaTario
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 degrees standard?

(https://en.wikipedia.org/wiki/Golden_ratio)

Best wishes,

DaTario

Last edited by a moderator:
DaTario said:
Hi All,

I have just found in the internet an identity showing that the Gonden Ration can be expressed as a function of the cossine 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 degrees standard?
Have you done ANY research to see where 360 degrees came from? It's not hard to do.

The 360 degrees came out of mathematicians using the sexagesimal number system:

https://en.wikipedia.org/wiki/Sexagesimal

Sexagesimal is great for doing fractional arithmetic because 60 can be factored in 2 * 3 * 2 * 5 which allows for fractionals like 1/2, 1/3, 1/5, 1/4, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30, 1/60 and any combination thereof.

phinds said:
Have you done ANY research to see where 360 degrees came from? It's not hard to do.

I have done some research and the sexagesimal system is somewhat familiar to me, its factorization properties, anatomic origin and etc. But one of the sentences the research offered me was:
"the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed." - Wikipedia

So I am asking.
Sorry if I disturbed you.
Simple suggestion for you not to be further disturbed with my questions is to disregard them in a next oportunity.

Best wishes,

DaTario

36 degrees are just 1/10 of a circle. If that would have played any role in choosing the number of degrees in a circle, 10 would have been a much more natural choice. But I don't see any indication that the golden ratio would have played a role. The sexagesimal system is much older than trigonometry.

mfb said:
36 degrees are just 1/10 of a circle. If that would have played any role in choosing the number of degrees in a circle, 10 would have been a much more natural choice. But I don't see any indication that the golden ratio would have played a role. The sexagesimal system is much older than trigonometry.
Ok, but not necessarily trigonometry was the body of knowledge used. Natural proportions (possibly anatomic), I would say, may have been the crucial knowledge to this connection, as they were known since long.

Besides, the role played by number of days in a year seems, imo, to add more confusion to this discussion.

Thank you mfb for your contribution.
Best wishes,

DaTario

## 1. What is the significance of 360 degrees in relation to the Golden Ratio?

The number 360 is traditionally used to represent a full rotation or circle in geometry. The Golden Ratio, also known as phi (φ), is an irrational number that is approximately equal to 1.618. This number has been found to have many mathematical and aesthetic properties, including its appearance in the geometry of a regular pentagon and the spiral growth patterns of certain plants and animals. The number 360 is often used in conjunction with the Golden Ratio to represent a full rotation while also incorporating the concept of balance and proportion.

## 2. How is the Golden Ratio used in art and design?

The Golden Ratio has been used by artists and designers for centuries as a principle of aesthetic harmony. It can be seen in the proportions of famous works of art, such as the Mona Lisa and the Parthenon, as well as in modern designs for architecture, furniture, and graphic design. The ratio is believed to create a sense of balance and beauty in art and design, and many believe it is naturally pleasing to the human eye.

## 3. Can the Golden Ratio be found in nature?

Yes, the Golden Ratio can be found in many aspects of nature, from the spirals of seashells and the branching of trees to the proportions of our bodies and even the arrangement of our internal organs. This ratio is believed to be a fundamental building block of the universe and can be seen in the growth patterns of many living organisms.

## 4. How is the Golden Ratio related to the Fibonacci sequence?

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. This sequence is closely related to the Golden Ratio, as the ratio between any two consecutive numbers in the sequence gets closer and closer to φ (1.618) as the sequence goes on. The Fibonacci sequence can also be seen in the spirals found in certain plants, shells, and galaxies, which follow the proportions of the Golden Ratio.

## 5. Are there any practical applications of the Golden Ratio?

While the Golden Ratio is primarily used in art and design, it has also been applied in fields such as mathematics, architecture, and even stock market analysis. Some believe that incorporating the Golden Ratio into design can create more visually appealing and harmonious structures, while others see potential in using the ratio for data analysis and prediction. However, the practical applications of the Golden Ratio are still a topic of debate among scientists and mathematicians.

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