Fibonacci Sequence: Lesson 6

 

Geometric Constructions and the Golden Ratio

You’ve seen the properties of the golden rectangle. 

 

And we got this value for ϕ from solving the proportion.

 

How do you construct a golden rectangle? It’s surprisingly simple and the steps in construction will also allow us to create other golden ratio polygons.

Constructing a Golden Rectangle

For this activity you can use a pencil, compass, and ruler on plain or grid paper. You can also do this activity using geometry software.

Step 1: Unit Square

Start by constructing a unit square, which measures 1 unit on each side. 

 

Step 2: Find the Midpoint of the Base

Use the compass and ruler to find the perpendicular bisector of the base of the square. 

 

Step 3: Construct a Segment from the Midpoint to a Vertex

Construct the segment shown.

 

 

Also, make a note of the length of this segment. We can calculate it using the Pythagorean theorem.

Step 4: Construct a Circular Arc

Using the segment you constructed as a radius, draw a circular arc that extends beyond the rectangle, as shown.

 

 

Step 5: Construct the Rectangular Extension

Using the circular arc as a guide, extend the rectangle. 

 

Step 6: Measure the Length of the Rectangle

Using what you know about the length of the original square and the rectangular extension, find the length of the rectangular base. As you can see it is phi, which means that this is a golden rectangle.

 

 

Constructing a Golden Triangle

Now let’s look at what a golden triangle would look like. As with the golden rectangle, the ratio of two sides will need to be phi. Here is what the golden triangle looks like. 

 

Because this is an isosceles triangle, the base angles and the opposite sides are congruent.

 

Another property of isosceles angles has to do with the vertex angle.

 

Using the sine ratio, we get this equation:

We can plug in different values for a and b that are in the ratio of phi, or close to it. What numbers that we’ve studied have this property? The Fibonacci numbers of course. 

 

1, 1, 2, 3, 5, 8, ….

 

Let’s start with the first two numbers and see what we get for theta:

 

 

But we also know that as the values of the Fibonacci pairs increase, their ratios approach phi. We can set up a spreadsheet to look at increasing values of the Fibonacci pairs to see if the angle converges on a particular value. Here is the spreadsheet.

 

 

As you can see the value of theta converges on 18°, which means that our golden triangle looks like this.

 

 

This golden triangle is a prominent part of the five-sided pentagram.

 

 

The interior of the pentagram is a regular pentagon and the triangular shapes are all golden rectangles. You’ll also find another golden ratio in the pentagon itself.

 

Let’s first look at a regular pentagon and one of its diagonals, as shown here.

 

Next, let’s inscribe a pentagram in the pentagon and adjust the angles, as shown here.

 

 

Knowing what you’ve learned about the golden triangle, can you see how the green and red segments below form a golden ratio?

 

 

Try to identify other golden ratios in this illustration.