The polar equation of a logarithmic spiral curve is given by
$$ r= ae^{b\theta} $$
where each point on the curve is described in polar coordinates: $r$ is the distance from the origin and $\theta$ is the angle formed with the x-axis. The parameters $a$ and $b$ are arbitrary.
Here's a description* for a construction scheme of the curve with varying precision (dependent on the number of rays):
The logarithmic spiral can be constructed from equally spaced rays by
starting at a point along one ray, and drawing the perpendicular to a
neighboring ray. As the number of rays approaches infinity, the
sequence of segments approaches the smooth logarithmic spiral (Hilton
et al. 1997, pp. 2-3).
Below are some examples* using different number of rays:
Questions:
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Why is it called a logarithmic spiral? Is it because the distance between neighbouring points on the x-axis happen to grow logarithmically?
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Apparently, there is a connection between the logarithmic spiral and the golden ratio. But where does this ratio occur? I reckon it's a limiting length for infinitely many rays, but I fail to see the connection both graphically and algebraically (i.e., can we predict the appearance of a golden ratio using solely the equation?)
References:
Best Answer
All spirals of the form $r=e^{b\theta}$ are logarithmic spirals. In complex form this would be expressed as $z=e^{(b+i)\theta}$. The parameter $b$ is called the flair coefficient, defined as $b=\ln g/\Delta\theta$, where $g$ is the growth rate of the radius per $\Delta\theta$. The golden spiral is specifically defined by $b=\ln \varphi/(\pi/2)=2\ln \varphi/\pi$, where $\varphi$ is the golden ratio. This means that the radius grows by a factor $\varphi$ for each quarter turn ($90^{\circ}$).