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

5 January 2019 The Primes and Factors page has been updated. It computes primes, factors and provides examples of Goldbach's conjecture in its "strong" form (from 4 on, every even number is a sum of 2 odd primes) and "weak" form (every odd number from 9 onwards is a sum of 3 primes)

... and to start your new year, a joke from Facebook's Easy Beautiful Maths group Happy New Year!

Fibonacci Numbers and Golden sections in Nature

listen again

• 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

• 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 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?

The Golden Section

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

Hear the whole programme

• The Golden section - the Number
  The golden section is also called the golden ratio, the golden mean and Phi.
  
  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.
  • 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.
  • 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

• 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 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 350 formulae and equations showing the properties of the Fibonacci and Lucas series, the general Fibonacci G series and Phi also available as
• Linear Recurrence Relations and Generating Functions
  An experimental page to list as many (linear) reurrence relations as I can find on ALL of the topics on these pages: Common Number series, Fibonacci Numbers and their many Generalisations, Pythagorean Triangles, Continued Fractions, Polygonal Numbers (ordinary and central), Egyptian Fractions.
• Links and Bibliography
  Links to other sites on Fibonacci numbers and the Golden section together with references to books and articles.

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Hosted by the Department of Mathematics, Surrey University, Guildford, UK, where the author was a Lecturer in the Mathematics and Computing departments 1979-1998.

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