Unravelling the Fibonacci Sequence
1,1,2,3,5,8,13,21… Do you see these numbers anywhere around you? Look more carefully because they are everywhere! If you are holding this magazine you can probably see this pattern out of the corner of your eye! Confused? Read on to clear your thoughts.
The Basics
This pattern of numbers is called the Fibonacci sequence. The Fibonacci numbers were first recognised as maatraameru (mountain of cadence) by the Sanskrit grammarian Pingala. But, the sequence is named after Leonardo de Pisa, also known as Leonardo Fibonacci, the greatest European mathematician of the middle ages. Leonardo Fibonacci was the first European to relate this sequence of numbers to the natural world around him. This sequence: 1,1,2,3,5,8,13,21,34… adds two sequential numbers together to get the next.
For example 1+1=2, 1+2=3, 2+3=5, and this continues infinitely. This sequence could be developed by anybody who knows how to add, then why is it so special? It is special because there are innumerable examples in nature, including honeybee populations and flower petal arrangements, which exhibit the Fibonacci sequence or Fibonacci numbers. The bones in your hand are a prefect example.
Bee hive order
The honeybee family tree gives a perfectly accurate depiction of the Fibonacci sequence. The queen bee of the hive lays eggs. Those that are fertilised by a male bee develop into a female and those that are not are male. Thus, male bees have only one parent. The family tree (previous page) with the youngest generation at the bottom will give you a better idea. Each male bee in the hive has 1 parent, 2 grandparents, 3 great grandparents, 5 great great grandparents, and so on. Now you get it?
Patterned Petals
Even if you look at a flower and count its petals, it will be a number in the sequence. This lily appears to have six petals, however, only the top three are petals and the bottom three are sepals. And 3 is a Fibonacci number. Even their seed heads follow the sequence!
Another pattern
There is another interesting pattern formed by the Fibonacci sequence: the Fibonacci rectangle. Each side of each box has a length equal to a Fibonacci number and when put into a spiral, each box length is the sum of the last two, the Fibonacci sequence. When a quarter circle, with a radius of the box’s length, is drawn in each box a spiral is formed. This spiral is very similar to Nautilus Shell.
A closer look
Hemachandra, a Jain philosopher, and Gopala, an Indian mathematician, did studies of these numbers. Hemachendra studied the Sanskrit vowel sounds that are either long (L) or short (S), and computed how many cadences of total length could be developed if the long is twice the length of the short. What he discovered was the Fibonacci sequence.
1 |
mora: S (forms only one pattern) |
1 |
2 |
morae: SS, L |
2 |
3 |
morae: SSS, SL, LS |
3 |
4 |
morae: SSSS, SSL, LSS, SLS, LL |
5 |
5 |
morae: SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS |
8 |
The studies done by Fibonacci initiated new observations in nature and the Golden Mean. Johannes Kepler showed that if a Fibonacci number is divided by the Fibonacci number that precedes it in the sequence, the ratios after 1 divided by 1 settle around one number, the golden mean, 1.618033…
1/1=1, 2/1=2, 3/2=1.5, 5/3=1.6666667, 8/5=1.6, 13/8=1.625 and so on.
Using Fibonacci
Many scientists, including Leonardo DaVinci, have studied this mean. It is used in art, music, architecture, and has opened a whole new field of scientific drawing and the study of the human anatomy. The Golden Mean is also called the Divine Proportion, which has been used to describe the proportions of the human body. Leonardo DaVinci used the Divine Proportion in his artwork to accurately draw human features. His drawing of the Vitruvian Man was the first of such accuracy.
Do you know that the distance from your wrist to your elbow is the length of your foot? Five times the width of your eyes is same as the size of your face! There is one eye length between your eyes! And your wingspan is your height! Can you now see these numbers around you?