Other Maths Pages at this site:

Pythagorean triangles
Right-angled triangles with integer sides, e.g. 3, 4, 5.
Exact Trig Values for Simple Angles updated
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.
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 new!
Fraction to Decimal with fixed and recurring parts in full, 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!.
Numbers which are the sum of a run of consecutive whole numbers
More on Runsums
The number 2016
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
19 April 2017
A workaround to the Mac iOS 10.3 Javascript bug which caused the new Calculator to hang (see 13 March entry below) has been installed :-)
25 March 2017
The new Calculator (see next item) now includes in inverse trig functions (arcsin, arccos and arctan) and inverse hyperbolic functions (arcsinh, arccosh, arctanh).

The Exact Trig Values for Simple Angles now includes a new section on how to use the new Calculator (mentioned in the 13 March 2017 item) for evaluating trig functions.

13 March 2017
Updates continue on various pages and there is a revised section on reverse and other transforms of continued fractions here

There is a new Decimal↔Fraction↔Continued Fraction Calculator, completely revamped to use very long decimal numbers, exact decimal fractions including the recurring part, CF terms and a periodic part together with an Expression Evaluator with as many decimal places as you might wish.
The old Fraction Calculator and Continued Fraction Calculator will remain for now but will eventually be replaced by this new and extended version.

The University of Surrey Open Days in 2017 will again host Dr Ron Knott's popular Mini-Maths talks on:
Friday 30 June,
Saturday 1 July,
Saturday 16 September,
Saturday 14 October
These are free 15-20 minute "fun" talks on maths suitable for a general audience and held in the Mathematics Department on top floor of the AA building: see this PDF map of the campus.
On each day Ron will be giving three different presentations at 11:45, 12:30 and 14:15.
For more details of how to get to the University of Surrey in Guildford, here is some travel information for train, coach and car. Everyone welcome!

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


Awards for this WWW site

Each icon is a link to lists of other Award winning sites that opens in a new window. Check them out!
Intute: formerly EEVL www.enc.org Games and fun for everyone MERLOT classic Award 2003 Eisenhower National ClearingHouse: www.enc.org Exploratorium Ten Cool Sites: www.exploratorium.edu/learning_studio/sciencesites.html 'Surfing the Net with Kids' 5 star site. www.surfnetkids.com www.britannica.com/ Scout Report: scout.cs.wisc.edu/report/sci-eng/current/index.html Science Master: www.sciencemaster.com Education Planet: www.educationplanet.com  www.schoolsnet.com  www.bigchalk.com www.coolmath.com www.yahbooks.com scientium.com/drmatrix Knot of Braids
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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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