## Fibonacci Numbers and the Golden Section

**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 The |

**There is a large amount of information here on the Fibonacci Numbers and related series and the on the Golden section, so if all you want is a quick introduction** then the first link takes you to **an introductory page** on the Fibonacci numbers and where they appear in Nature.

The rest of this page is a brief introduction to all the web pages at this site onFibonacci Numbers the Golden Section and the Golden Stringtogether with .their many applications |

What’s New? – the FIBLOG The 15th Fibonacci Numbers and Applications conference at Eger, Hungary, June 25-30 2012 with guest speaker Neil Sloane. If you are nearby, come and say Hi! 19 June 2012:Fib and Phi Formulae updated and corrected |

## 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 listen againonline or download the podcast. It is a useful general introduction to the Fibonacci Numbers and the Golden Section.

- 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. - The Golden section in Nature

Continuing the theme of the first page but with specific reference to**why**the golden section appears in nature. Now with a Geometer’s Sketchpad dynamic demonstration.0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

### 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 Easier Fibonacci Puzzles page

has the 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! - The Harder Fibonacci Puzzles page

still has problems where the Fibonacci numbers are the answers – well, all but ONE, but WHICH one? If you know the Fibonacci Jigsaw puzzle where rearranging the 4 wedge-shaped pieces makes an additional square appear, did you know the*same*puzzle can be rearranged to make a different shape where a square now*disappears*?

For these puzzles, I do not know of any simple explanations of*why*the Fibonacci numbers occur – and**that’s the real puzzle – can you supply a****simple**reason why??

### The Intriguing Mathematical World of Fibonacci and Phi

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

- The Mathematical Magic of the Fibonacci numbers

looks at the patterns in the Fibonacci numbers themselves: the Fibonacci numbers in Pascal’s Triangle; using the Fibonacci series to generate all right-angled triangles with integers sides based on Pythagoras Theorem.- An auxiliary page:
- More on Pythagorean triangles

If you want to look like a number wizard to your friends then try out the simple Fibonacci numbers trick!

The following pages give you lots of opportunities to find your own patterns in the Fibonacci numbers. We start with a complete list of…- The first 500 Fibonacci numbers…

completely factorized up to Fib(300) and all the prime Fibonacci numbers are identified up to Fib(500). - A Formula for the Fibonacci numbers

Is there a direct formula to compute Fib(n) just from n? Yes there is! This page shows several and why they involve Phi and phi – the golden section numbers. - Fibonacci bases and other ways of representing integers

We use base 10 (decimal) for written numbers but computers use base 2 (binary). What happens if we use the Fibonacci numbers as the column headers?

- An auxiliary page:

### 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..

## which we call Phi (note the capital **P**) 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**) 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.

Hear the whole programme (14 minutes) using the free RealOne Player.

- The Golden section and Geometry

The golden section is also called the golden ratio, the golden mean and the divine proportion.Two more pages look at its applications in Geometry: first in flat (or two dimensional) geometry and then in the solid geometry of three dimensions.

- Two-dimensional Geometry and the Golden section or Fantastic Flat Facts about Phi

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 where we phind phi phrequently!- An auxiliary page on Exact Trig Values for Simple Angles explores the many places that Phi and phi occur when we try to find the exact values of the sines, cosines and tangents of simple angles like 36° and 54°.

- The Golden Geometry of Solids or Phi in 3 dimensions

The golden section occurs in the most symmetrical of all the three-dimensional solids – the Platonic solids. What are the best shapes for fair dice? Why are there only 5?

The next pages are about the numbers Phi = 1·61803.. and phi = 0·61803… and their properties.

- Phi’s Fascinating Figures – the Golden Section number

All the powers of Phi are just whole multiples of itself plus another whole number. Did you guess that these multiples*and*the whole numbers are, of course, the Fibonacci numbers again? Each power of Phi is the sum of the previous two – just like the Fibonacci numbers too.- Introduction to Continued Fractions

is an optional page that expands on the idea of a continued fraction (CF) introduced in the Phi’s Fascinating Figures page. - There is also a Continued Fractions Converter (a web page – needs no downloads or special plug-is) to change decimal values, fractions and square-roots into and from CFs.
- This page links to another auxiliary page on Simple Exact Trig values such as cos(60°)=1/2 and finds all simple angles with an exact trig expression, many of which involve Phi and phi.

- Introduction to Continued Fractions
- Phigits and Base Phi Representations

We have seen that using a*base*of the Fibonacci Numbers we can represent all integers in a binary-like way. Here we show 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).

- Two-dimensional Geometry and the Golden section or Fantastic Flat Facts about Phi

### 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 Rabbit Sequence

See show how the golden string arises directly from the Rabbit problem and also is used by computers when they compute the Fibonacci numbers. You can hear the Golden sequence as a sound track too.

The Fibonacci Rabbit sequence is an example of a**fractal**– a mathematical object that contains the whole of itself within itself infinitely many times over.

### Fibonacci – the Man and His Times

- Who was Fibonacci?

Here is a brief biography of Fibonacci and his historical achievements in mathematics, and how he helped Europe replace the Roman numeral system with the “algorithms” that we use today.

Also there is a guide to some memorials to Fibonacci to see in Pisa, Italy.

### More Applications of Fibonacci Numbers and Phi

- 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 and discover its properties.

It ends with a number trick which you can use “to impress your friends with your amazing calculating abilities” as the adverts say. It uses facts about the golden section and its relationship with the Fibonacci and Lucas numbers.- The first 200 Lucas numbers and their factors

together with some suggestions for investigations you can do.

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ….

- The first 200 Lucas numbers and their factors
- The Fibonomials

The basic relationship defining the Fibonacci numbers is F(n) = F(n – 1) + F(n – 2) where we use some combination of the previous numbers (here, the previous two) to find the next. Is there such a relationship between the*squares of the Fibonacci numbers*F(n)^{2}? or the*cubes*F(n)^{3}? or other powers? Yes there is and it involves a triangular table of numbers with similar properties to Pascal’s Triangle and the binomial numbers: the Fibonomials. - General Fibonacci Series

The Lucas numbers change the two starting values of the Fibonacci series from 0 and 1 to 2 and 1. What if we changed these to any two values? These General Fibonacci series are called the G series but*the*Fibonacci series and Phi again play a prominent role in their mathematical properties. Also we look at two special arrays (tables) of numbers, the Wythoff array and the Stolarsky array and show how a these two collections of general Fibonacci series contain*each whole number exactly once*. The secret behind such clever arrays is … the golden section number Phi!

### Fibonacci and Phi in the Arts

- Fibonacci Numbers and The Golden Section In Art, Architecture and Music

The golden section has been used in many designs, from the ancient**Parthenon**in Athens (400BC) to Stradivari’s violins. It was known to artists such as**Leonardo da Vinci**and musicians and composers, notably**Bartók**and**Debussy**. This is a different kind of page to those above, being concerned with speculations about where Fibonacci numbers and the golden section both do and do not occur in art, architecture and music. All the other pages are factual and verifiable – the material here is a often a matter of opinion. What do*you*think?

### Reference

- Fibonacci and Phi Formulae

A reference page of about 300 formulae and equations showing the properties of the Fibonacci and Lucas series, the general Fibonacci G series and Phi. Also available in PDF format (21 pages) for which you will need the free Acrobat PDF Reader or plug-in for your browser. - Links and Bibliography

Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles.