Fibonacci Sequence: Lesson 3

 

The Fibonacci Sequence

The Fibonacci sequence started with rabbits, led to a nonlinear number pattern, and left us with an interesting visualization. In the previous lesson we saw how a visualization of the Fibonacci sequence produced this interesting pattern.

 

There is a curve that spirals out of this visual pattern.

This curve is known as a logarithmic spiral and the Fibonacci  sequence is connected to these spirals. In this lesson we explore the properties of these spirals.

The Archimedean Spiral

Spiral shapes can be created using a polar coordinate system. The simplest such spiral is called the Archimedean spiral. This is the graph of it. Note the polar equation.

The Archimedean spiral is NOT a logarithmic spiral. It is made of up of this locus of points: Imagine a point on a circle that is expanding in size and rotating. This animation shows how that point behaves and how its path defines the Archimedean spiral:

The Arhimedean spiral is based on two key ideas: 

• A circle is rotating at a constant speed.
• The radius of the circle is growing at the same speed.

A point on the circle defines the shape of the spiral. This is why the equation is r = θ. Equal changes in the rotation (θ) results in equal changes in the radius (r). 

Notice that the Archimedean spiral is more circular than a logarithmic spiral and doesn’t expand as dramatically as the logarithmic spiral does. In fact, this graph has an equivalent in the Cartesian system, y = x. Every corresponding change in x results in a corresponding change in y.

This is a linear graph. Even though it doesn’t look linear, the Archimedean spiral is also a linear graph. All this means is that equal changes in r result in equal changes in θ. But we know that the Fibonacci sequence is nonlinear. In fact it looks like exponential growth.

The Logarithmic Spiral

We can conclude the logarithmic spiral is exponential. So let’s now explore what an exponential polar graph looks like.

Now you can see the logarithmic spiral having the same shape as the logarithmic spiral. For a tighter spiral, include a coefficient less than 1 in the exponential term. Here’s an example.

How does the logarithmic spiral differ from the Archimedean spiral? This animation shows that for a logarithmic spiral, where the expanding circle intersects the spiral the angle formed by the line tangent to the circle and the tangent to the spiral is constant. You can see the constant angle in this animation.

Here’s another way at looking at the logarithmic spiral shape. At every point on the logarithmic spiral, the angle formed by the tangent to the curve and a radial line forms a constant angle, alpha. See the animation below.

This constant angle formed means that the logarithmic spiral is self-similar. What this means is that as the spiral grows, it still has the same shape. 

There’s another geometric shape that has this self-similar property. A fractal is a self-similar object.

Notice the logarithmic spiral in this fractal image. The logarithmic spiral occurs a lot in nature. See the logarithmic spirals in these illustrations?

Why does this shape occur so often in nature, especially among living things? The answer is simple: Most organisms grow in size but they maintain their shape. This is an example of self-similarity.

Children are usually smaller versions of their parents. The small-size, same-shape phenomenon is captured by the Fibonacci sequence and the logarithmic spiral.