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Old 2020-01-14, 12:38   #1
Jul 2014

3×149 Posts
Default problem to do with golden ratio equation

Can anyone explain what's wrong with my logic?
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Old 2020-01-14, 13:09   #2
axn's Avatar
Jun 2003

5,179 Posts

Golden ratio is an increasing ratio (i.e > 1). The first equation uses x as a decreasing ratio (i.e. x < 1). So you get 1/gr when you solve that.
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Old 2020-01-14, 14:45   #3
Dr Sardonicus
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Feb 2017

53×41 Posts

The usual formulation for x and y being in golden proportion is \frac{x}{y}\;=\;\frac{x\;+\;y}{x}; the right-hand side clearly is greater than 1. Taking y = 1 gives

\frac{x}{1}\;=\;\frac{x\;+\;1}{x}\text{, or }x^{2}\;-\;x\;-\;1\;=\;0\text{.}

An illustration is given by the 72-72-36 degree isosceles triangle. The bisector of one of the 72-degree angles divides the opposite side in golden ratio; calling x the length of the base and y the length of the smaller segment of the side opposite the angle bisector, gives the above proportion.
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Old 2020-01-14, 14:54   #4
Jul 2014

44710 Posts

Thanks very much to both of you.
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