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you add all the sides

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Because it's pleasing to the eye and is widely used by painters because of its mysterious beauty.

Its sides are in the golden ratio of 1:(1+sq rt of 5)/2 or about 1:1.618

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Q: Why do we appreciate the golden rectangle?
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Can the golden rectangle formula be used on squares?

No not normally


How do get 14 as answer for whole number pairs of side lengths rectangle that forms golden rectangle?

A golden rectangle cannot have both its sides as whole numbers. The ratio of the sides of the rectangle is [1 + sqrt(5)]/2 so if one side is a positive whole number, the other must be an irrational number.


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