The Fibonacci Numbers and the Golden section

University Crest University Crest Department of Computing
School of Electronic Engineering, IT and Mathematics
University of Surrey
Guildford, Surrey, UK

Fibonacci Numbers and the Golden Section

What's On These Pages

This is the University of Surrey Home page for the Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13,...
(which one is next?) and the golden section and related numbers:
0.61803 39887... or 1.61803 39887...
as well as a sequence of 0s and 1s called the golden string
1 0 1 1 0 1 0 1 1 0 1 1 0 ..

The Fibonacci pages at this site:

This page has been kept relatively free of images for faster downloading.
The pages linked to below have more images.
Who was Fibonacci?
A brief biography of Fibonacci and his achievements in mathematics and memorials to Fibonacci to see in Pisa, Italy.
Fibonacci Numbers and Nature
Fibonacci and the original problem about rabbits where the series first appears, the family trees of cows and bees, the golden ratio and the Fibonacci series, the Fibonacci Spiral and sea shell shapes, branching plants, flower petal and seeds, leaves and petal arrangements, on pineapples and in apples, pine cones and leaf arrangements. All involve the Fibonacci numbers - and here's how and why.
More on the Fibonacci series and related mathematics:
The first 100 Fibonacci numbers and their factors
101st-300th Fibonacci numbers and 301st-500th Fibonacci numbers.
Mathematical Patterns in the Fibonacci numbers
looks at the patterns in the Fibonacci numbers themselves, the Fibonacci numbers in Pascal's Triangle.
There are many investigations for you to do to find patterns for yourself.
A Formula for the Fibonacci numbers
Is there a direct formula to compute Fib(n) just from n? Yes there is! This page shows what it is and why it involves Phi and phi - the golden section numbers.
The Golden Section
is also sometimes called the golden ratio or the golden mean or the divine proportion and is often denoted by the greek letter phi which is sometimes printed as or by another greek letter tau . It is closely connected with the Fibonacci series and has a value of 1.61803... . We call it Phi on these pages, give its value to 2000 decimal places. It has some interesting properties. Phi is (sqrt(5)+1)/2 and phi is (sqrt(5)-1)/2 .
Two pages are devoted to the marvellous geometrical facts about the golden section number - first in flat (or two dimensional) geometry and then the solid geometry of three dimensions.
Fantastic Flat Phi Facts
See some of the unexpected places that the golden section (Phi) occurs in Geometry and in Trigonometry: pentagons and decagons, paper folding and Penrose Tilings.
The trigonometry section looks at the graphs of the sine, cosine and tangent functions and Phinds Phi Phrequently!
The Golden Geometry of the Solid Section or Phi in 3 dimensions
The golden section occurs in the most symmetrical of all the three-dimensional globe-like solids - the Platonic solids. What are the best shapes for fair dice? Why are there only 5?
Phi's Fascinating Figures - the Golden Section number
All the powers of Phi are just whole multiples of itself plus another whole integer (and guess what these whole integers are? Of course - the Fibonacci numbers again!)
Here you can see some of the amazing numerical properties of Phi and the Fibonacci numbers.
Introduction to Continued Fractions page takes these ideas further.
Some puzzles involving the Fibonacci Numbers
See Fibonacci numbers in brick wall patterns, Fibonacci bee lines, seating people in a row and the Fibonacci numbers again, giving change and a game with match sticks and even with electrical resistance and lots more puzzles all involve the Fibonacci numbers!
It's all on this page!
If you know the Fibonacci Jigsaw puzzle where rearranging the pieces makes a additional square appear, did you know the same puzzle can be rearranged to make a another shape where a square now disappears?
Fibonacci Rabbit Sequence
There is another way to look at Fibonacci's Rabbits problem that gives an infinitely long sequence of 1s and 0s, which we will call the Fibonacci Rabbit sequence:-
1 0 1 1 0 1 0 1 1 0 1 1 0 ...
which is a close relative of the number the golden section and the Fibonacci numbers. You can hear the Golden sequence as a Quicktime movie track too!
The Fibonacci Rabbit sequence is an example of a fractal.
The Golden Section In Art, Architecture and Music
The number Phi occurs has been used to design many buildings from the ancient Parthenon in Athens (400BC) to Le Corbusier's United Nations building in New York. It was well known by artists such as Leonardo da Vinci and musicians too from composers like Bartok to the famous violin maker Stradivari.
Fibonacci bases and other ways of representing integers
You know we use base 10 for written numbers, and computers use base 2, but what about using the Fibonacci numbers as the column headers? or base Phi (1.618034...) instead of base 10?
Phigits and Base Phi Representations
is an advanced page if you have mastered all the above. It shows that there is an interesting way of representing all integers in a binary-like fashion but using only powers of Phi instead of powers of 2 (binary) or 10 (decimal).
The Fibonacci numbers in a formula for Pi ().
There are several ways to compute pi (3.14159 26535 ..) accurately. One that has been used a lot is based on a nice formula for calculating which angle has a given tangent, discovered by James Gregory. His formula together with the Fibonacci numbers can be used to compute pi. This page introduces you to all these concepts from scratch.
Fibonacci Forgeries
Sometimes we find series that for quite a few terms look exactly like the Fibonacci numbers, but, when we look a bit more closely, they aren't - they are Fibonacci Forgeries.
Since we would not be telling the truth if we said they were the Fibonacci numbers, perhaps we should call them Fibonacci Fibs !!
The Lucas Numbers
Here is a series that is very similar to the Fibonacci series, the Lucas series, but it starts with 2 and 1 instead of Fibonacci's 0 and 1. It sometimes pops up in the pages above so here we investigate it some more - or rather, there are lots of interesting and simple investigations for you to do to discover its properties for yourself!
New results added to the Things To Do section (March 98).
The first 100 Lucas numbers and their factors
together with some suggestions for investigations you can do.
Links and references
There are many links on the pages above as well as references to some excellent books and articles.
This page contains links to other sites on Fibonacci numbers and the Golden section so there is more yet for you to explore.
I am preparting a list of Fibonacci and Phi Formulae. An incomplete draft version (March 98) is available as a 1.6Mb postscript file (.ps) which is gnu-zipped (.gz) down to 802Kb. This will be expanding shortly as more fomulae are added.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Awards for this WWW site

Alive! Excellence in Education Eisenhower National ClearingHouse StudyWeb Award Exploratorium Ten Cool Sites "Surfing the Net with Kids" 5 star site Britannica Internet Guide
Each icon is a link to lists of other Award winning sites - check them out!
StudyWeb has given Awards to three pages at this site in January 1998: The Fibonacci numbers in a formula for Pi, The Fibonacci numbers and Nature and Introduction to Continued Fractions.

A talk at your school/college/organisation?

Would you like a talk at your school, college or other organisation on the Fibonacci Numbers and the Golden Section? I have given such talks at the Orkney Science Festival (the second largest Science Festival in the UK in a wonderful part of the world - the Orkney Islands just off the north coast of Scotland) and also at many schools and colleges. Click on the "Dr Ron Knott" link at the foot of the page for contact details.
This page has been accessed times since March 1996.
Currently there are around 200 visitors per day to this page.
The counter passed 80,000 in May 1998, so if it is around zero then our Web server disk has become full again which seems to clear this counter to zero!
It'll be restored fairly soon!

Brought to you from the Department of Computing of Surrey University in Guildford in the county of Surrey in the UK.
Dr Ron Knott (email:
last updated 19 June 1998