Other Maths Pages at this site:

Triangles and Geometry
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?
A Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates
introduction to trilinear and barycentric coordinates, links to Clark Kimberling's Encyclopedia of Triangle Centers (ETC) with automatic lookup
Primes & Factors Calculator
Integer Bases Updated!
many kinds of bases to represent integers
Integer Palindromes
Polygonal and Figurate Numbers
Triangular, Square, Pentagonal ... Numbers
More on Polygonal Numbers
Central polygonal numbers, matchstick figures, 3D solid shapes and higher dimensions
Numbers which are the sum of a run of consecutive whole numbers
More on Runsums
Integer Sums or Partitions of an integer
The number 2016
The number 2021
Fractions and Decimals
- their periods and patterns and in non-decimal bases.
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.
An Exact Fractions Calculator and converter
Fractions to and from Decimal, Continued Fractions from and to Fractions with an expression evaluator and many in-built functions and all to as many decimal places as you like!.
Linear Recurrence Relations and Generating Functions
Lock and Roll!
A Yahtzee-type dice game
Got it!
A Countdown-type numbers game,

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 = 1 + phi = 1/phi

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.

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.

What's New? - the FIBLOG
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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 (podcast, 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 went live in March 1996 and is therefore the oldest maths site on the web!
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!
© 1996-2018 Dr Ron Knott ronknott at mac dot com Built With BBEdit
Created (March) 1996,