Fibonacci Numbers and the Golden Section

This is the Home page for the Fibonacci numbers, the Golden section and the Golden string.

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... and ±1·61803 39887...

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 which is closely related to the Fibonacci numbers and the golden section.

There is a large amount of information at this site (more than 200 pages if it was printed), 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 on
Fibonacci Numbers the Golden Section and the Golden String
together with their many applications.

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What's New? 7 June 2001
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Fibonacci Numbers and Golden sections in Nature

* 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. New!Now with a Geometer's Sketchpad dynamic demonstration.

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 Phi and phi phi.
* The Mathematical Magic of the Fibonacci numbers
looks at the patterns in the Fibonacci numbers themselves, the Fibonacci numbers in Pascal's Triangle and using Fibonacci series to generate all right-angled triangles with integers sides based on Pythagoras Theorem.
Impress your friends with a simple Fibonacci numbers trick!
There are many investigations for you to do to find patterns for yourself as well as a complete list of...
+ The first 500 Fibonacci numbers...
completely factorised up to Fib(300) and all the prime Fibonacci numbers are identified.
+ 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?
* The Golden Section - the Number and Its Geometry
The golden section is also called the golden ratio, the golden mean and the divine proportion. It is closely connected with the Fibonacci series and has a value of (sqrt5 - 1)/2 which is 0·61803... which we call phi on these pages. It has some interesting properties such as 1/phi is the same as 1+phi and we call this value Phi= (sqrt5 + 1)/2.

Two pages are devoted to its applications in Geometry - first in flat (or two dimensional) geometry and then in 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 where we phind 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 solids - the Platonic solids. What are the best shapes for fair dice? Why are there only 5?

The next pages are about the number Phi = 1·61803.. itself and its close cousin phi = 0·61803... .

+ 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.
o Introduction to Continued Fractions An optional page that expands on the idea of a continued fraction introduced in the Phi's Fascinating Figures page.
+ 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).

The Golden String

The golden string also referred to as the Infinite Fibonacci Word or the Fibonacci Rabbit sequence.
* 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 1 0 1 1 0 1 ...

which is a close relative of 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.

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 ( 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 100 Lucas numbers and their factors
together with some suggestions for investigations you can do.
* 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 page to those above, being concerned with speculations about where the golden section both does and does 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 - but interesting nevertheless!

Links and References

* Fibonacci, Phi and Lucas numbers Formulae
A reference page of over 100 formulae and equations showing the properties of these series.
New! Now available in PDF format (96K) for which you will need the free Acrobat PDF Reader or plug-in.
* Links and references
Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles.

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

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The Knot a Braid of Links Project at Camel designated this page a cool math site of the week for 22-28 November 1998 (now available via in the Kabol Database search engine).


This site is listed in the BBC Education Web Guide (January 1999).

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