The Fibonacci Sequence and the Golden Ratio

The famous Fibonacci sequence is the infinite, ordered list of numbers: 1, 1, 2, 3, 5, 8, 13, 21, ... It is full of countless, intriguing patterns that may be explored by students young and old. If you have not seen it before, try to figure out how to predict the next number in the sequence!

The picture below is based on the Fibonacci sequence. If you don't see the pattern in the numbers, the picture may help (if you look at it carefully for a while)! I like to share this picture with students after they have learned about the Fibonacci sequence, but I don't tell them about the connection. I let them discover that for themselves. A couple of the  "noticing and wondering" items below talk of something called the golden ratio. There is more information about this at the bottom of the page.

I notice that the picture is composed of squares of different sizes.
I notice that the squares combine to form (non-square) rectangles.
I notice a pattern beginning with two small red squares inside the picture.
I wonder where I should draw the next larger square in the pattern.
I notice that each square is attached to the the longer side of a smaller rectangle.
I notice that larger squares are attached in a spiraling pattern: above, right, below, left...

I notice that the side lengths of the squares are the Fibonacci numbers.
I notice that the rectangles look like they might be similar.
I notice that the 2 by 1 rectangle is not similar to the 3 by 2 rectangle.
I notice that the large:small ratio of the side lengths is getting closer and closer to the golden ratio.

I wonder what the areas of the rectangles are.
I wonder if the ratios of the rectangles' areas are getting closer to some number.
I wonder if this number is related to the golden ratio.
I wonder how I can use what I know about the connections between scale factors and area ratios to understand these patterns.

I wonder what would happen to the picture and to all of these patterns if I began the picture with three small red squares instead of two!

The golden ratio, ø, (phi) is approximately 1.618033989. Like π (pi), it is an irrational number— a number that cannot be expressed exactly as a repeating or terminating decimal (or as a simple fraction). The ratios of neighboring Fibonacci numbers gradually approach the golden ratio. This means that the large:small ratios in the picture above get ever closer to ø and, consequently, that the successive rectangles get closer to being similar as they grow larger. Students may discover that the ratios of the areas approach ø-squared, which turns out to be the same value as ø + 1. (This is related to the fact that the area ratio of similar triangles is always the square of the length ratio.)

Algebra students may explore further in many directions. How may the golden ratio be expressed using radicals? How about the square of the golden ratio? Can you find a simple relationship between ø and its reciprocal? Can you write the areas of each of the rectangles in the picture as linear expressions in terms of ø? What patterns do you see in the coefficients of these expressions?