The Golden Ratio and Fibonacci Numbers in Nature

By: Mary Catherine Clark

Leonardo Fibonacci was the most outstanding mathematician of the European Middle Ages.

He was known by other names including Leonardo Pisano or Leonard of Pisa.

Little was know about his life except for the few facts given in his mathematical writings.

Fibonacci was born around 1170. Received his early education from a

Muslim schoolmaster. His first book was published in 1202

called Liber Abaci. This book is devoted to arithmetic and elementary algebra.

His next book, Practica Geometriae, he wrote in 1220. This book presents geometry and trigonometry with Euclidean originality.

He employed algebra to solve geometric problems and geometry to solve algebraic problems. This was radical in Europe at the time.

He wrote two other books. One of which included Liber Quadratorum. This earned him the reputation as a major number theorist.

He was one of the first people to introduce the Hindu-Arabic number system to Europe- the system used today.

Liber Abaci remained the European standard for more than two centuries replacing the Roman numeration system.

Today a statue of Fibonacci stands in a garden across the Arno River, near the Leaning Tower of Pisa.

The sequence in which each number is the sum of the two preceding numbers.

This sequence is defined by the linear recurrence equation.

and by the definition above  The Fibonacci numbers for n=1,2 are 1, 1,

2, 3, 5, 8, 13, 21,… Before Fibonacci wrote his work, these

numbers had already been investigated by Indian Scholars interested in rhythmic patterns of syllables.

Fibonacci’s book Liber Abcai asked a question involving the reproduction of a single pair of rabbits which is the basis of the Fibonacci sequence.

It was posed: Suppose a newly born pair of rabbits (a male and female) are put in a field. The rabbits are able to mate at the age of one month so that at the end of the second month a female can produce another pair of rabbits. Assuming that the rabbits never die and the female always produces a new pair every month from the second month on. How many pairs will there be in one year?

Answer: 144 rabbits

The golden ratio is an irrational number defined to be

This has a value of 1.61803 and is sometimes denoted by φ after the mathematician Phidias who studied its properties.

Its unique properties were first considered in the idea of dividing a line into two segments such that the ratio of the total length to the length of the longer segment is equal to the ratio of the length of the longer segment to the length of the shorter segment.

The golden rectangle can be constructed from these line segment so that the length to width ratio is φ.

The golden rectangle may be divided into a square and a smaller golden rectangle.

The ancient Greeks believe that a rectangle constructed in this manner was the most aesthetically pleasing of all rectangles and they incorporated this shape into a lot of their art and architectural designs.

The ratio of successive Fibonacci numbers is something you might be surprised by!

As n increases, the ratio of approaches the golden ratio and is expressed as =

This is the fundamental property of both the Fibonacci sequence and the golden ratio.

Both of these ratios converge at the same limit and are the positive root of the quadratic equation

If the two smallest squares have a width and height of 1, then the box to their left has a measurement of 2 and the other boxes measure 3, 5, 8, and 13.

The golden ratio is expressed in spiraling shells.

There is a quarter of a circle in each square going from one corner to the opposite.

This is not a true mathematical spiral.

Look at any seed head, and you will notice what look like spiral patterns curving out form the center left and right.

If you count these spirals you will find a Fibonacci number. If you look at the spirals to the left and then the right you will notices these are two consecutive Fibonacci numbers.

These can also be seen in pinecones, pineapples, cauliflower, and much more!

Most of the time, the number of pedals on a flower is a Fibonacci number!

1 pedal-calla lily2 pedals-euphorbia

3 pedals-trillium

5 pedals-columbine

8 pedals-bloodroot

13 pedals-black eyed susan

Dunlap, Richard A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.

Koshy, Thomas. Fibonacci and Lucas Numbers with Applications. New York: John Wiley & Sons, inc., 2001.

Vorobiev, Nicolai N. Fibonacci Numbers. 6th. Basel: Birkhauser Verlag, 1992.

Http://www.world-mysteries.com/sci_17.htm

http://science.howstuffworks.com/evolution/fibonacci-nature1.htm

http://mathworld.wolfram.com/FibonacciNumber.html

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html

http://www.branta.connectfree.co.uk/fibonacci.htm