The Golden Ratio

An irrational constant (like Euler's number e, and pi - 3.14159265...) that occurs frequently in nature, the Golden Ratio (also known the golden mean, golden section) is best summarised by stating that the lengths a and b are to a, as a is to b (see diagramme below).

Symbolised by the Greek lower case letter 'phi', the ratio is also observed in the Fibonacci sequence, a sequence that is produced when one takes two terms that are then added together to produce the next term within, beginning at one (i.e. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...). The significance of this ratio is that it can be, and is, found in the most unexpected of places (ex. a flower will often have a Fibonacci number of petals), and perhaps underlies the very nature of reality itself. In the clip below, it's mentioned that the relationship between the Earth and the moon is defined by it (at the approx. 15:06 mark), and as we all know, during an eclipse of the sun by the moon, the moon from our perspective on Earth appears to be the exact same size as the sun, covering it perfectly and leaving only the corona visible.

This video isn't as long, and explains it very well.

The Golden Ratio
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• Here's one for ya, put a circle inside any triangle formed by side lengths of a pythagorean triple and tell me the radius

would you have expected the result?

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• forgot to add "... of any pythagorean triple such that the circle is tangent to all sides and tell me the radius..."

• The length of the radius (and hence size of the circle) would depend upon the particular Pythagorean Triple that is used.

• yes that's true, however that's not what's important. Is there anything significant about a circle that is tangent to all sides of a triangle formed by a pythagorean triple? Is it odd that it's consistent for all other pythagorean triples?

• Fascinating stuff. The world's full of funky shit, but the only golden ratio I care about is that 0.6-0.7 waist:hip ratio

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• It's pretty much just a way of framing things. That's why it works for almost everything. Part of it gives a square and a rectangle and repeating the process and joining the curve makes a circular shape. So that basically covers almost everything.