21.5 Logarithmic spirals

Before you read this, I suggest you read post 21.3.

In post 21.3 we met the simple spiral whose equation in polar coordinates is

r = aθ                    (1)

where a is a constant, r is the radius of the spiral and θ is an angle that is positive for anticlockwise rotation.

In this post we are going to look at the logarithmic spiral whose equation is

r = r0eaθ.                    (2)

Here r0 is the radius when θ = 0 and e is the irrational number that has a value of about 2.718. The picture at the beginning of this post shows a logarithmic spiral with r0 = 1 and a = 0.002. It resembles the spiral in the picture of the shell below.

Why is it called a logarithmic spiral when there are no logarithms in equation 2? I suppose its because we could define

ρ = loger = loger0 + aθ.

For a given spiral r0 is constant, so changing its value simply moves the origin of the Cartesian coordinate system in the first picture. If we move the origin so that loger0 =0 we obtain the result that

ρ = aθ

which has the same form as equation 1 but now ρ is the logarithm of the radius of the spiral.

The golden spiral is defined to have

a = (2logeϕ)/π

when θ is measured in radians. Here ϕ is an irrational number called the golden ratio that has a value of about 1.618, so now a ≈ 0.306 or 0.0053 if θ is measured in degrees. It is sometimes claimed that a marine animal called nautilus (see picture above) has a shell that is a special shape because it is a logarithmic spiral. But any logarithmic spiral with a value of a that is roughly that for a golden spiral will look like a nautilus shell. I constructed the logarithmic spiral below in degrees with a = 0,005. If you look at enough shells, it’s not surprising that you’ll find one that looks like a golden spiral.

Now let’s think what happens if θ increases with time; θ = ωt where ω is the angular speed. Then

r = r0eaωt = r0ekt

where k is a constant for a constant value of ω. This equation represents exponential growth of the radius. We can then think of the spiral of the nautilus, and other, shells, as being the consequence of exponential growth. The growth of bacteria shows us that numbers of biological cells grow exponentially if they have an adequate food supply. So, it’s not surprising if the radius of a shell grows exponentially.

Related posts

20.3 Polar coordinates, circles and spirals
18.15 More about exponential growth

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