Other Maths Pages at this site:

The number 2016 New!
Pythagorean triangles
Right-angled triangles with integer sides, e.g. 3, 4, 5.
Exact Trig Values for Simple Angles
Which angles have a simple exact value for their sine,cosine or tangent?
Fractions and Decimals
All about decimals and fractions and their periods and patterns and other bases than 10.
Fractions Calculator
Convert fractions to and from decimal fractions, find their exact decimal fraction and repeating parts
Farey Fractions and Stern-Brocot Tree Calculators
Two ways of arranging all fractions
Egyptian fractions
The Egyptians only had unit fractions of the form 1/n. How did they use them?
Introduction to Continued Fractions
An unusual method of writing fractions that has many advantages.
Numbers which are the sum of a run of consecutive whole numbers
More on Runsums
Polygonal and Figurate Numbers
Linear Recurrence Relations and Generating Functions

Looking for something specific on these Maths pages?

Fibonacci Numbers and the Golden Section

This is the Home page for Dr Ron Knott's multimedia web site on the Fibonacci numbers, the Golden section and the Golden string hosted by the Mathematics Department of the University of Surrey, UK.

The Fibonacci numbers are
0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next)

The golden section numbers are

0·61803 39887... = phi = φ and
1·61803 39887... = Phi = Φ

The golden string is

1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...
a sequence of 0s and 1s that is closely related to the Fibonacci numbers and the golden section.

What's New? - the FIBLOG
25 September 2016
New formulae added to the Fibonacci and Phi Formulae page.
Autumn 2016
Gradually all pages are being updated to use new logos/icons for calculators and bibliographies and an automatically generated Contents section, so all pages will use JavaScript and this need to be enabled in your browser.
Updates to the Maths of the Fibonacci Numbers page
and a new section Every number starts some Fibonacci number
New version of the Fibonacci Calculator:
The Fibonacci Numbers Calculator version 3 now deals with even larger numbers and for any General Fibonacci series where the Fibonacci Rule (add the previous two numbers to find the next) applies from any two given starting values. Also computes Pisano periods and Entry pointsand plots barcharts of Benford's Law on initial digits.
Version 3.1 has a narrower Calculator that fits better on mobile phones and tablets in portrait mode.

If you want a quick introduction then have a look at the first link on the Fibonacci numbers and where they appear in Nature.

THIS PAGE is the Menu page linking to other pages at this site on the Fibonacci numbers and related topics above.

Fibonacci Numbers and Golden sections in Nature

Ron Knott was on Melvyn Bragg's In Our Time on BBC Radio 4, November 29, 2007 when we discussed The Fibonacci Numbers (45 minutes). You can Radiolisten again online or download the podcast. It is a useful general introduction to the Fibonacci Numbers and the Golden Section.

The Puzzling World of Fibonacci Numbers

A pair of pages with plenty of playful problems to perplex the professional and the part-time puzzler!

The Intriguing Mathematical World of Fibonacci and Phi

The golden section numbers are also written using the Greek letters Phi Phi and phi phi.

The Golden Section

The golden section number is closely connected with the Fibonacci series and has a value of (√5 + 1)/2 or:

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More.. Calculator

which we call Phi (note the capital P), the Greek letter Φ, on these pages. The other number also called the golden section is Phi-1 or 0·61803... with exactly the same decimal fraction part as Phi. This value we call phi (with a small p), the Greek letter φ here. Phi and phi have some interesting and unique properties such as 1/phi is the same as 1+phi=Phi.
The third of Simon Singh's Five Numbers programmes broadcast on 13 March 2002 on BBC Radio 4 was all about the Golden Ratio. It is an excellent introduction to the golden section. I spoke on it about the occurrence in nature of the golden section and also the Change Puzzle.
RadioHear the whole programme (14 minutes) using the free RealOne Player.

The Golden String

The golden string is also called the Infinite Fibonacci Word or the Fibonacci Rabbit sequence. There is another way to look at Fibonacci's Rabbits problem that gives an infinitely long sequence of 1s and 0s called the Golden String:-
1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 ...

This string is a closely related to the golden section and the Fibonacci numbers.

Fibonacci - the Man and His Times

More Applications of Fibonacci Numbers and Phi

Fibonacci and Phi in the Arts


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This site first appeared on the web in March 1996
Hosted by the Department of Mathematics, Surrey University, Guildford, UK, where the author was a Lecturer in the Mathematics and Computing departments 1979-1998.

Blast From the Past: An archived snapshot of this site as it was in June 1998 and at various times from 1999 to 2005 from www. archive.org!
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