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.
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.
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.
The Puzzling World of Fibonacci Numbers
A pair of pages with plenty of playful problems to perplex the
professional and the part-time puzzler!
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!
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
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...
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.
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
(5 - 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=
(5 + 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.
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 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... .
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
An optional page that expands on the idea of a continued fraction introduced in
the Phi's Fascinating Figures page.
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.
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.
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.
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
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!!
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 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!
A reference page of over 100 formulae and equations showing the properties
of these series. Now available in
PDF format (96K) for which you will need the
free Acrobat PDF Reader or plug-in.
7 June 2001
Fibonacci Numbers and Nature
Now with a Geometer's Sketchpad dynamic
demonstration.
and phi
. 
The first 500 Fibonacci numbers...
5 - 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=
(
Introduction to Continued Fractions
An optional page that expands on the idea of a continued fraction introduced in
the Phi's Fascinating Figures page.
)
!!
