Title: The Fibonacci Series and The Golden Ratio
1The Fibonacci Series and The Golden Ratio
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987,
Adam Rowlands Grey College
2Main Points
- Background
- Brief mathematical properties
- So whys this interesting?
3Background
What is the Fibonacci series?
- A simple numerical series devised by the Italian
mathematician Leonardo de Pisa towards the end of
the 12th century - It was derived from the rabbit problem and was
described in his most famous book - Liber Abaci
(Book of the Abacus) - A pair of rabbits produces a pair of baby
rabbits once each month. Each pair of baby
rabbits requires one month to grow to be adults
and subsequently produces one pair of baby
rabbits each month thereafter. Determine the
number of pairs of adult rabbits some number of
months after - He discovered that the problem could be modelled
mathematically using the below additive sequence - An2 An An1
-
4How was this equation derived?
- The number of adult rabbits in month n2 is the
combined number of adults and babies from the
previous month - An2 An1 bn1
- In a given month (say n1) the number of baby
rabbits is equal to the number of adult rabbits
in the previous month - bn1 An
- Hence An2 An An1
- Numerous sequences can be produced using the
above equation, but the most famous The
Fibonacci series, uses the root values 0 and 1 - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987,
5Finding a fibonacci number
- Trivial method of adding previous terms
- A more sophisticated method involves the number
Phi (t) - Fn 1/t n (-t) - n
- Binet formula - Jacques Phillipe
- There are two problems with this method which
cannot be overcome - Phi is an irrational number
- If n is very large and Fn is required to be a
precise integer then Phi must have a sufficient
number of significant digits - Hence Fn trunct n /5 0.5
6Generalised Fibonacci Numbers
- The most famous series produced using the same
equation but different root values is the Lucas
series - 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,
322, 521, 843
- A specific Lucas number can be found using a
similar equation to the Binet formula
- The Lucas numbers are related to the Fibonacci
numbers - Ln v5Fn
7How does this relate to Phi?
- The fundamental link between the Fibonacci series
and the number phi - t limn?8 Fn / Fn-1
8Mathematical properties of Phi
- An irrational number defined as (1v5)/2
- Phi 1.61803
- No pleasing symmetry
- Phi appears throughout nature and art
- t v(1 t)
- Thus
- t v(1 v(1 v(1 v(1 v(1 v(1 ..))
9Where does Phi occur?
- The Great Pyramid of Cheops
- Nature - flowers and fruit e.g. pineapples,
Nautilus Shell
10The Great Pyramid of Cheops
11Examples of the Fibonacci series
- Nature
- Pinecones
- Sunflowers
- Petals on flowers
12How does this relate to teaching?
- Field work
- Different way of looking at mathematics
- Historical viewpoint
- Mystery surrounding numbers
- Studying famous mathematicians
13Summary
- The Fibonacci series and phi are some of the
oldest concepts in mathematics - They have been surrounded by intrigue and mystery
for centuries e.g. Leonardo Da Vinci - They exhibit numerous unexpected relations with
various aspects of mathematics and the natural
world - Possible applications to teaching
140, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987,
The Fibonacci Series and The Golden Ratio
Adam Rowlands Grey College