The Golden Ratio

Image courtesy of wikipedia.org

The Golden Ratio is typically represented by the Greek letter Phi and is an irrational number that has found its place in nature. As illustrated in the diagram below, most plants follow a similar when sprouting a new section:

Image courtesy of unitone.org

If we multiply 1 over Phi squared by the 360 degrees in a circle we will get the 137.5 degree of separation that most plant employ. This angle is continued from the last sprouted section as the plant continues to grow upward and in doing so creates a spiral type pattern. This precise pattern produces the lease amount of overlap with respect to the plants leaves and therefore is the most beneficial for the collection of sunlight by the plant.
The human body also exemplifies the fascinating occurrence of Phi in the natural world, for example the bones in our fingers are related to each other by a ratio of Phi:

Image courtesy of unitone.org

The Golden Ratio can be matched to many patterns in nature including the nautilus shell mentioned above. The Fibonacci squares used above can also form Golden Rectangles, which are rectangles that have Phi as the ratio between the length and width. The incredible aspect of a Golden Rectangle is that a square can be cut from the rectangle and a smaller Golden Rectangle will remain. This process can be repeated over and over to create an ever-shrinking Golden Rectangle:

Image courtesy of mathforum.org

The relationship between Fibonacci numbers and the Golden Ratio is simple. The higher you increase in the sequence the closer the difference between the current number and the next numbers is to the Golden Ratio. For example, in the Fibonacci sequence the ratio between 5 and 8 is 1.6, while the ratio between two sequential numbers higher in the scale such as 679891637638612258 and 1100087778366101931 is 1.6180339887, which is much closer to the Golden Ratio.