I can construct a golden spiral by rotating segments 90 degrees and scaling their segments by the golden ratio, and then use the polar formula on wikipedia to interpolate points at other angles to smooth it out.

However, I do not see a formula for other metallic spirals like the silver spiral. Can I assume the same process? What is the polar equation for a silver spiral?


The equation for the golden spiral is given by $$ {\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}$$ with $$\varphi = \frac{1+\sqrt{5}}{2}$$ the golden ratio.

Analogously, the equation for the silver spiral is given by $$ {\displaystyle r=\delta_S ^{\theta {\frac {2}{\pi }}}\,}$$ with $$\delta_S = 1 + \sqrt{2}$$ the silver ratio.

Note that in general the equation $$r = x ^{\theta {\frac {2}{\pi }}}$$ leads to a ratio $x$ for the radii differing by a quarter turn.


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