Notes from

1. The Fibonacci Sequence

What is the relationship between the following numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...

2. The Golden Ratio

Take the ratio of two successive numbers in Fibonacci's series and divide each by the number before it. What number is being approached in this series?

This number is called Phi.

3. The Golden Rectangle/ The Fibonacci Spiral

Take a piece of paper and draw two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1).

Draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units).


You should end up with a drawing that looks like this:

If you draw a half circle through each of the squares, you wind up with a spiral that looks like this:


4. The Fibonacci Spiral in Nature

 Once the Greeks had discovered this intriguing number, they began to notice that it showed up in various forms in nature:

Many seed heads on plants have patterns which appear to be the Fibonacci Spiral:

Look at this pinecone:

The leaf arrangements of common plants follow the spiral as well:

You even see it in cauliflower:

Here you can see the same spiral in the shape of a snail:

When the Greeks noticed the connection between the Fibonacci sequence and many organic (living) forms, they believed that they had discovered a divine number.

What would Plato have argued in a discussion of the Fibonacci Sequence?

5. The Golden Rectangle in Architecture:

The Greeks celebrated the Fibonacci sequence in their art and architecture:

Check out the Parthenon. See the Fibonacci Spiral?


Look again: (animation)

Artists throughout history have followed the Greek interest in the connections between numbers and nature: