Logarithmic Spirals

Logarithmic Spirals
Fields: Geometry and Algebra
Image Created By: Unknown

Logarithmic Spirals

Logarithmic spirals are spirals which appear in nature, such as in this nautilus shell. They possess the remarkable property that the distances between the turnings are in a geometric progression.

Basic Description

To understand the logarithmic spiral, we will first examine the spiral itself. We remove the axes and add concentric circles. The smallest circle is of radius 1 and every other circle has a radius of 1 greater than the previous circle's.

If we look at where the spiral intersects the the horizontal axis on the right, we notice that the spiral intersects the horizontal axis at the circle of radius 1, 3, and 9 - a geometric sequence.

We must also see that each intersection occurs when the spiral has complete one full revolution, or rotated 2 $\pi$ radians. By increasing at a fixed rate, the angles of intersection are a arithmetic sequence.

Thus we see the defining feature of logarithmic spirals: the radii increases geometrically while angle increases arithmetically.

Places In Nature

Logarithmic spirals are approximated by the shape of galaxies, nautilus shells(as pictured in the main image), and hurricanes.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Algebraic Analysis

We can see this property algebraically when we examine the function form that [...]

[Click to hide A More Mathematical Explanation]

Algebraic Analysis

We can see this property algebraically when we examine the function form that determines the spirals.

Logarithmic spirals are graphs of the form

$r = ae^{b\theta}\,$

or inversely,

$\theta = \frac{1}{b} \ln(r/a)\,$

To simplify calculations we will use the radian form, with a = 1, b= 1.

Thus $r = e^{\theta}\,$.

To find several r values, we pick different values of theta.

$r_1 = e^{0} = 1\,$

$r_1 = e^{\pi/2} = 1 * e^{\pi/2} = {r_0}*(e^{\pi/2})\,$

$r_2 = e^{\pi} = 1 * {e^{\pi/2}}*e^{\pi/2} = {r_0}*(e^{\pi/2})^2\,$

$r_3 = e^{\frac{3\pi}{2}} = 1 * {e^{\pi/2}}*e^{\pi/2}*e^{\pi/2} = {r_0}*(e^{\pi/2})^3\,$

We can see that the radius increases by a factor of $e^{\pi/2}\,$ for every increase in $\frac{\pi}{2}$.

It is important to note that the spiral's radii increase geometrically for any constant change in angle and conversely, the spiral's angle in. In this example we arbitrarily picked a constant increase of $\frac{\pi}{2}$ to complement the picture, but we would have found a geometric series regardless of the angle we choose to increase.

Golden Spiral

A golden spiral is a special logarithmic spiral that is formed when the ratio of radii is in the golden ratio every 90 degrees.

The equation is of the form

$r = e^{b\theta}\,$

$|b| = {\ln{\phi} \over 90} = 0.0053468\,$ for $\theta$ in degrees;

$|b| = {\ln{\phi} \over \pi/2} = 0.306349\,$ for $\theta$ in radians.

Some spirals approximate the golden spiral, such as the one pictured below.

If we look at each square, we notice that each contains a quarter circle. Each square's side are in a ratio of the $\varphi$.

However, this is not a golden spiral because it was constructed in this manner; the radii themselves are constant within the squares. An actual golden spiral has radii that are constantly changing.

Other Properties

Self-Similiarity

From the animation, it is apparent that the spiral has the same shape as we zoom in.

In this animation, the upper bound is $4 \pi$ and the lower bound is $-12 \pi$. Therefore this graph's similarity would end once the graph ended. However, if we set the lower limit of $\theta$ to $-\infty$, we will get infinite self-similarity, and by definition, a fractal.

Finite Length

If we allow the $\theta$ value to approach $-\infty$, the spiral will continue indefinitely. However, it is interesting to note the while the spiral continues infinitely, it has a finite length.

We can deduce this from using calculus but here we will present another more intuitive way.

Imagine a logarithmic spiral that begins at some fixed radius a and whose lower radius bound is approaching 0. This spiral lies on a flat plane with a 3 dimensional space.

Now we take this spiral out of it original plane and position it up right, with the point at which it has radius a is tangent to the original plane. We now unfurl it.

As we have seen before, the turnings of the spirals are in geometric sequences. If we choose to cut the spiral at these specific turnings we get an infinite number of line segments whose lengths are a decreasing geometric sequence.

We can sum these line segment lengths by using the formula, which clearly gives us a finite number.

Ideas for the Future

- Make a smoother animation for the self similar section.

- Somehow highlight the logarithmic spiral in the nautilus shell in the main image.

- Definitely need a animation of my intuitive way of seeing how it has finite length.

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