Constructing a Golden Spiral

Notice how we built our rectangle in a counterclockwise direction. This leads us into another interesting characteristic of the Golden Ratio. Let's look at the rectangle with all of our construction lines drawn in:

We are going to concentrate on the squares that we drew, starting with the two smallest ones. Let's start with the one on the right. Connect the upper right corner to the lower left corner with an arc that is one fourth of a circle:

Then continue your line into the second square on the left, again with an arc that is one fourth of a circle:

We will continue this process until each square has an arc inside of it, with all of them connected as a continuous line. The line should look like a spiral when we are done. Here is an example of what your spiral should look like (move your cursor over the images to see them change):

Now what was the point of that? The point is that this "golden spiral" occurs frequently in nature. If you look closely enough, you might find a golden spiral in the head of a daisy, in a pinecone, in sunflowers, or in a nautilus shell that you might find on a beach. Here are some examples: