15 February 2022 To mark the Chinese New Year a new section on
The Remainders Number System
has been added to the Number Bases page, based on the Chinese Remainder Theorem.
2 February 2022
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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 (45 minutes).
listen again online
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
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
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?
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
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.
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
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.
Hear the whole programme
(14 minutes) using the free
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 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
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
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.
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
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 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?
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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This site went live in March 1996 and is therefore the oldest maths site on the web!
Hosted by the
Department of Mathematics,
where the author was a Lecturer in the Mathematics and Computing departments 1979-1998.