Investigating the Fibonacci Series

One of the topics that has been enjoyed was 'Sequences and Series' – the study of number patterns.
 
The boys and girls of Set A launched an investigation into one of the most interesting of sequences, the Fibonacci Series, and were amazed by what they discovered.

Leonardo Fibonacci – by Max Bruneau and James Connolly
Leonardo Fibonacci was born in the year 1175. He was the son of a wealthy merchant called Guglielmo Bonacci.
 
He was also known as Leonardo of Pisa, Leonardo Pisano Bigollo, Leonardo Pisano, Leonardo Bonacci but most commonly as simply Fibonacci.
 
He was considered the most talented mathematician in Europe during the Middle Ages.
 
Fibonacci is best known for spreading the Hindu-Arabic number system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc) that we use today. This was because he found it easier to use when adding, subtracting, multiplying and dividing than the more complicated Roman Numerals.
 
Fibonacci is also famous for his discovery of the Fibonacci Sequence.
 
Fibonacci died in the year 1250, aged 75 – a ripe old age for the time!

 

The Fibonacci Sequence Explained – by Hubie Litherland and Evan Ball

The Fibonacci Sequence was discovered by Leonardo Fibonacci in 1202.
 
It is an ingenious piece of maths which works by adding the previous two numbers in the series together.
 
This is an example of how it works:
 
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 488... etc
 
The Fibonacci Sequence can be found throughout Nature e.g. the shell of the nautilus, flower heads and pine cones…..even cauliflowers. As the numbers in the series get higher and higher the ratio between each number and the number preceding it begin to form something called the “Golden Ratio”.
 

Parthenon

The Parthenon, Athens, Greece

 

The height and width of Classical Greek buildings like the Parthenon, the lengths of the sides of an A4 piece of paper – even the dimensions of Isabella Ramsay’s face are all related to one another by the Golden Ratio.

 

Fibonacci Spirals – by Ella Robson and Ruby Stuart

You can make a basic Fibonacci Spiral of your own. If you take an A4 sheet of paper and fold it carefully, then fold it again and again as many times as possible before drawing a smooth curved line through the edge of each box formed, you will create a spriral as below:

Spiral

Fibonacci/Golden Spiral

Fibonacci Spirals can be found throughout Nature. It is thought that they develop due to naturally occurring lines that develop as a plant, animal or even galaxy grows. Here are a few examples of where Fibonacci Spirals can be found:
  • Flower heads
  • Nautilus shells
  • Pinecones
  • Fruits
  • Vegetables
  • Hurricanes
  • Galaxies
  • Sea shells
  • Ocean waves
The area of each box will be in direct relation to its place in the Fibonacci Sequence.
 
You can see how similar the nautilus shell below is to this pattern:

Nautilus Shell

Nautilus Shell

 

The seeds of this sunflower head are arranged in Fibonacci Spirals. There are 8 clockwise spirals and 13 anti-clockwise spirals: both are Fibonacci numbers:

Sunflower Head

Sunflower Head

 
 This amazing Romanesque cauliflower is literally covered in Fibonacci Spirals!

Romanesque Cauliflower

Romanesque Cauliflower - looks amazing; smells horrible!

 

A Bee’s Family Tree – by Annabel Barlow, Isabella Ramsay and David Wakefield

Honey Bee

Honey Bee

 
Fibonacci famously discovered that the family tree of any honey bee fits the Fibonacci Sequence exactly. Bees are either male or female. The males are called drones and have two parents, a male and a female (the queen bee). A female bee has only one parent – the queen bee.
 
The family tree of any honey bee therefore looks like this:

Family Tree

Honey Bee Family Tree

 
If you add up the number of bees in each row (with the queen at the bottom) you can see that it exactly matches the Fibonacci Sequence – 1, 1, 2, 3, 5 etc – AMAZING!!
 

Conclusion

All in all this has been the most amazing voyage of discovery for the pupils of Form 4.
 
This investigation clearly demonstrates that there is real magic and mystery to be found out there in the wild world of Mathematics!