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Can we possibly go back to the definition of the “golden ratio” [1]?
In the golden ratio article on Wikipedia, the term ‘golden ratio’ is defined:
The golden ratio, golden mean, golden section, etc., are synonyms for the Greek letter phi (φ), derived from the first two letters of the Greek words phi-losophia, meaning “philosophy, love of wisdom”, and the second two letters of phi-sophia, meaning “loving wisdom”. The term itself is Greek and is φυσικός, meaning “physical” in Greek.
But why is the ‘golden ratio’ defined this way? Why not take an arbitrary ratio of some kind, and say it is the golden ratio, like this:
The golden ratio, golden mean, golden section, etc., are synonyms for a ratio with certain properties. That ratio is the golden ratio.
[1] ‘Like we can define the case in which the ratio is a third of the value of one of the numbers, in the same way we can define the cases when the ratio is a fifth of the value of one of the numbers.’
A:
It’s the ratio which contains the property. For example, the square root is a ratio, but the golden ratio is not a ratio; if you construct the golden ratio by a repeated addition of phi, then, if the ‘golden addition’ method is continued, you get the area of the regular pentagon.
A:
I agree with @Gerald Edgar’s answer, but to answer the actual question:
Why is the ‘golden ratio’ defined this way?
It’s not. The definition on Wikipedia is incorrect. But it is a natural way of talking about ratios if you define phi to mean “any ratio at all”.
Why not take an arbitrary ratio of some kind, and say it is the golden ratio
Because there is only one natural way to talk about ratios, and that is using the letter phi.
It is confusing to have many different natural ways of talking about ratios, and it is particularly confusing to use multiple different words for the same concept.
A:
In any particular construction of a
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