Why does an equiangular spiral become logarithmic (intuitively)?

One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all the time while plotting it.

My question
I can see it in the picture that the spiral arms are getting bigger each turn and I see the math. What I don't understand intuitively is where the logarithm comes from - or put differently: Why does a geometric progression arise just by holding an angle constant. Could you give me some hints? Thank you,

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1 Answer

Much is explained by looking at the polar equation of the spiral:

$$r=\exp(\theta\cot\alpha)$$

Here, $\alpha$ is the constant angle any tangent to the curve makes with the radius vector (a line segment joining the origin and the point of tangency). This explains the adjective equiangular (the verification of this property from the defining equation is left as an exercise). As an aside, insects flying towards a point light source like a candle or a light bulb follow the path of an equiangular spiral, since the usual strategy of an insect flying at the daytime to get their bearing is to fly at a constant angle from the sun's rays, and this strategy works against them when encountering man-made light.

Now, suppose we have an arithmetic progression of angles $\theta,\theta+\Delta\theta,\theta+2\Delta\theta,\dots$; if we get the corresponding values of the radius vector using the defining equation for the logarithmic spiral (geometrically speaking, this corresponds to a clockwise rotation by $0,\Delta\theta,2\Delta\theta,\dots$ radians), we get

$$\exp(\theta\cot\alpha),\exp((\theta+\Delta\theta)\cot\alpha),\exp((\theta+2\Delta\theta)\cot\alpha),\dots$$

which can be re-expressed as

$$\exp(\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha),\exp(\theta\cot\alpha)\cdot\exp(\Delta\theta\cot\alpha)^2,\dots$$

which as you can see is a geometric progression; that is to say, the logarithms of the members of this sequence form an arithmetic progression. This is where the logarithmic adjective arises from.

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M.: Thank you (also for the book tip!) To be honest with you: Now I get the part with the geometric progression - but I still don't see the connection between the const. angle and the logarithm :-( – vonjd Oct 25 '10 at 16:35

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@vonjd: I'd chalk it up as a coincidence... the spiral happens to both have the constant angle property and radius vectors in a geometric progression. One way to proceed would be to derive the equation of the equiangular spiral from one property and then use it to derive the other. – J. M. Oct 25 '10 at 22:27
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