The Fibonacci numbers are Nature's numbering system.
They appear everywhere in Nature, from the leaf arrangement in
plants, to the pattern of the florets of a flower, the bracts of a
pinecone, or the scales of a pineapple. The Fibonacci numbers are
therefore applicable to the growth of every living thing, including
a single cell, a grain of wheat, a hive of bees, and even all of
Golden Ratio & Golden Section : : Golden Rectangle : :
Golden Ratio & Golden Section
In mathematics and the arts, two quantities are in the golden
ratio if the ratio between the sum of those quantities and the
larger one is the same as the ratio between the larger one and the
The golden ratio is often denoted by the Greek letter
phi (Φ or φ).
The figure of a golden section illustrates
the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant,
A golden rectangle is a rectangle whose side lengths are
in the golden ratio, 1: j
that is, 1 : or approximately 1:1.618.
A golden rectangle can be constructed with only
and compass by this technique:
Construct a simple square
Draw a line from the midpoint of one side of the square to
an opposite corner
Use that line as the radius to draw an arc that defines the
height of the rectangle
Complete the golden rectangle
In geometry, a golden spiral is a logarithmic spiral
whose growth factor b is related to j,
the golden ratio. Specifically, a golden spiral gets wider (or
further from its origin) by a factor of j
for every quarter turn it makes.
Successive points dividing a golden rectangle into squares lie
a logarithmic spiral which is sometimes known as the golden
Golden Ratio in Architecture and Art
Many architects and artists have proportioned their works
to approximate the golden ratio—especially in the form of the
golden rectangle, in which the ratio of the longer side to the
shorter is the golden ratio—believing this proportion to be
aesthetically pleasing. [Source: Wikipedia.org]
Here are few examples:
Parthenon, Acropolis, Athens.
This ancient temple fits almost precisely into a golden rectangle.
The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci
We can draw many lines of the rectangles into this figure.
Then, there are three distinct sets of Golden Rectangles:
Each one set for the head area, the torso, and the legs.
Image Source >>
Vetruvian Man is sometimes confused with principles of "golden rectangle",
however that is not the case. The construction of Vetruvian Man is
based on drawing a circle with its diameter equal to
diagonal of the square, moving it up so it would touch the base of
the square and drawing the final circle between the base of the
square and the mid-point between square's center and center of the
Detailed explanation about geometrical construction of the Vitruvian Man by Leonardo da Vinci >>
Golden Ratio in Nature
Adolf Zeising, whose main interests were mathematics and
philosophy, found the golden ratio expressed in the arrangement of
branches along the stems of plants and of veins in leaves. He
extended his research to the skeletons of animals and the
branchings of their veins and nerves, to the proportions of
chemical compounds and the geometry of crystals, even to the use
of proportion in artistic endeavors. In these phenomena he saw the
golden ratio operating as a universal law. Zeising wrote in
The Golden Ratio is a universal law in which is contained
the ground-principle of all formative striving for beauty and
completeness in the realms of both nature and art, and which
permeates, as a paramount spiritual ideal, all structures, forms
and proportions, whether cosmic or individual, organic or
inorganic, acoustic or optical; which finds its fullest
realization, however, in the human form.
Click on the picture for animation showing more examples of
A slice through a Nautilus shell reveals
golden spiral construction principle.
Fibonacci was known in his time and is still recognized today as
the "greatest European mathematician of the middle ages."
He was born in the 1170's and died in the 1240's and there is now a
statue commemorating him located at the Leaning Tower end of the
cemetery next to the Cathedral in Pisa. Fibonacci's name is also
perpetuated in two streetsthe quayside Lungarno Fibonacci in Pisa
and the Via Fibonacci in Florence.
His full name was Leonardo of Pisa, or Leonardo Pisano in Italian
since he was born in Pisa. He called himself Fibonacci which was
short for Filius Bonacci, standing for "son of Bonacci",
which was his father's name. Leonardo's father( Guglielmo Bonacci)
was a kind of customs officer in the North African town of Bugia,
now called Bougie. So Fibonacci grew up with a North African
education under the Moors and later travelled extensively around the
Mediterranean coast. He then met with many merchants and learned of
their systems of doing arithmetic. He soon realized the many
advantages of the "Hindu-Arabic" system over all the
others. He was one of the first people to introduce the Hindu-Arabic
number system into Europe-the system we now use today- based of ten
digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8
9. and 0
His book on how to do arithmetic in the decimal system, called Liber
abbaci (meaning Book of the Abacus or Book of calculating) completed
in 1202 persuaded many of the European mathematicians of his day to
use his "new" system. The book goes into detail (in Latin)
with the rules we all now learn in elementary school for adding,
subtracting, multiplying and dividing numbers altogether with many
problems to illustrate the methods in detail.
The sequence, in which each number is the sum of the two
preceding numbers is known as the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21,
34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, ...
(each number is the sum of the previous two).
ratio of successive pairs is so-called golden section
- 1.618033989 . . . . .
whose reciprocal is 0.618033989 . . . . . so
that we have 1/GS = 1 + GS.
The Fibonacci sequence, generated by the rule f1 = f2 = 1 , fn+1 = fn +
is well known in many
different areas of mathematics and science.
Pascal's Triangle and Fibonacci Numbers
The triangle was studied by B. Pascal, although it
had been described centuries earlier by Chinese mathematician
Yanghui (about 500 years earlier, in fact) and the Persian
astronomer-poet Omar Khayyám.
Pascal's Triangle is described by the following
is a binomial
The "shallow diagonals" of Pascal's
sum to Fibonacci numbers.
It is quite amazing that the Fibonacci number patterns occur so
frequently in nature
( flowers, shells, plants, leaves, to name a
few) that this phenomenon appears to be one of the principal "laws of nature".
Fibonacci sequences appear in biological settings, in two
consecutive Fibonacci numbers, such as branching in trees,
arrangement of leaves on a stem, the fruitlets of a pineapple, the
flowering of artichoke, an uncurling fern and the arrangement of a
pine cone. In addition, numerous claims of Fibonacci numbers or
golden sections in nature are found in popular sources, e.g.
relating to the breeding of rabbits, the spirals of shells, and the
curve of waves The Fibonacci numbers are also found in the
family tree of honeybees.
Fibonacci and Nature
Plants do not know about this sequence - they just grow in the most
efficient ways. Many plants show the Fibonacci numbers in the
arrangement of the leaves around the stem. Some pine cones and fir
cones also show the numbers, as do daisies and sunflowers.
Sunflowers can contain the number 89, or even 144. Many other
plants, such as succulents, also show the numbers. Some coniferous
trees show these numbers in the bumps on their trunks. And palm
trees show the numbers in the rings on their trunks.
Why do these arrangements occur? In the case of leaf arrangement,
or phyllotaxis, some of the cases may be related to maximizing the
space for each leaf, or the average amount of light falling on each
one. Even a tiny advantage would come to dominate, over many
generations. In the case of close-packed leaves in cabbages and
succulents the correct arrangement may be crucial for availability
of space. This is well described in several books listed here
So nature isn't trying to use the Fibonacci numbers: they are
appearing as a by-product of a deeper physical process. That is why
the spirals are imperfect.
The plant is responding to physical
constraints, not to a mathematical rule.
The basic idea is that the position of each new growth is about
222.5 degrees away from the previous one, because it provides, on
average, the maximum space for all the shoots. This angle is called
the golden angle, and it divides the complete 360 degree circle in
the golden section, 0.618033989 . . . .
Examples of the Fibonacci sequence in
Petals on flowers*
Probably most of us have never taken the time to examine very
carefully the number or arrangement of petals on a flower. If we were to
do so, we would find that the
number of petals on a flower, that still has all of its
petals intact and has not lost any, for many flowers is a Fibonacci
3 petals: lily, iris
- 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
- 8 petals: delphiniums
- 13 petals: ragwort, corn marigold, cineraria,
- 21 petals: aster, black-eyed susan, chicory
- 34 petals: plantain, pyrethrum
- 55, 89 petals: michaelmas daisies, the asteraceae family
Some species are very precise about the number of petals they
have - e.g. buttercups, but others have petals that are very near
those above, with the average being a Fibonacci number.
| One-petalled ...
|Two-petalled flowers are not common.
|Three petals are more common.
|Five petals - there are hundreds of species, both wild and cultivated, with five
|Eight-petalled flowers are not so common as five-petalled, but there
are quite a number of well-known species with eight.
|Twenty-one and thirty-four petals are also quite common. The outer
ring of ray florets in the daisy family illustrate the Fibonacci sequence
extremely well. Daisies with 13, 21, 34, 55 or 89 petals are quite common.
shasta daisy with 21
|Ordinary field daisies have 34
a fact to be taken in consideration when playing "she loves me,
she loves me not". In saying that daisies have 34 petals, one is
generalizing about the species - but any individual member of the species
may deviate from this general pattern. There is more likelihood of a
possible under development than over-development, so that 33 is more
common than 35.
* Read the entire article here:
Flower Patterns and Fibonacci Numbers
Why is it that the number of petals in a flower is often one of
the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the
lily has three petals, buttercups have five of them, the chicory has
21 of them, the daisy has often 34 or 55 petals, etc. Furthermore,
when one observes the heads of sunflowers, one notices two series of
curves, one winding in one sense and one in another; the number of
spirals not being the same in each sense. Why is the number of
spirals in general either 21 and 34, either 34 and 55, either 55 and
89, or 89 and 144? The same for pinecones : why do they have either
8 spirals from one side and 13 from the other, or either 5 spirals
from one side and 8 from the other? Finally, why is the number of
diagonals of a pineapple also 8 in one direction and 13 in the
© All rights reserved
Image Source >>
Are these numbers the product of chance? No! They all belong to
the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.
(where each number is obtained from the sum of the two preceding). A
more abstract way of putting it is that the Fibonacci numbers fn are
given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f
n+2 = fn+1 + fn . For a long time, it had been noticed that these
numbers were important in nature, but only relatively recently that
one understands why. It is a question of efficiency during the
growth process of plants.
The explanation is linked to another famous number, the golden
mean, itself intimately linked to the spiral form of certain types
of shell. Let's mention also that in the case of the sunflower, the
pineapple and of the pinecone, the correspondence with the Fibonacci
numbers is very exact, while in the case of the number of flower
petals, it is only verified on average (and in certain cases, the
number is doubled since the petals are arranged on two levels).
© All rights reserved.
Let's underline also that although Fibonacci historically
introduced these numbers in 1202 in attempting to model the growth
of populations of rabbits, this does not at all correspond to
reality! On the contrary, as we have just seen, his numbers play
really a fundamental role in the context of the growth of plants
THE EFFECTIVENESS OF THE GOLDEN MEAN
The explanation which follows is very succinct. For a much more
detailed explanation, with very interesting animations, see the web
site in the reference.
In many cases, the head of a flower is made up of small seeds
which are produced at the centre, and then migrate towards the
outside to fill eventually all the space (as for the sunflower but
on a much smaller level). Each new seed appears at a certain angle
in relation to the preceeding one. For example, if the angle is 90
degrees, that is 1/4 of a turn, the result after several generations
is that represented by figure 1.
Of course, this is not the most efficient way of filling space.
In fact, if the angle between the appearance of each seed is a
portion of a turn which corresponds to a simple fraction, 1/3, 1/4,
3/4, 2/5, 3/7, etc (that is a simple rational number), one always
obtains a series of straight lines. If one wants to avoid this
rectilinear pattern, it is necessary to choose a portion of the
circle which is an irrational number (or a nonsimple fraction). If
this latter is well approximated by a simple fraction, one obtains a
series of curved lines (spiral arms) which even then do not fill out
the space perfectly (figure 2).
In order to optimize the filling, it is necessary to choose the
most irrational number there is, that is to say, the one the least
well approximated by a fraction. This number is exactly the golden
mean. The corresponding angle, the golden angle, is 137.5 degrees.
(It is obtained by multiplying the non-whole part of the golden mean
by 360 degrees and, since one obtains an angle greater than 180
degrees, by taking its complement). With this angle, one obtains the
optimal filling, that is, the same spacing between all the seeds
This angle has to be chosen very precisely: variations of 1/10 of
a degree destroy completely the optimization. (In fig 2, the angle
is 137.6 degrees!) When the angle is exactly the golden mean, and
only this one, two families of spirals (one in each direction) are
then visible: their numbers correspond to the numerator and
denominator of one of the fractions which approximates the golden
mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc.
These numbers are precisely those of the Fibonacci sequence (the
bigger the numbers, the better the approximation) and the choice of
the fraction depends on the time laps between the appearance of each
of the seeds at the center of the flower.
This is why the number of spirals in the centers of sunflowers,
and in the centers of flowers in general, correspond to a Fibonacci
number. Moreover, generally the petals of flowers are formed at the
extremity of one of the families of spiral. This then is also why
the number of petals corresponds on average to a Fibonacci number.
An excellent Internet
site of Ron Knot's at the University of Surrey on this
and related topics.
S. Douady et Y. Couder, La physique des spirales
végétales, La Recherche, janvier 1993, p. 26 (In French).
Source of the above segment:
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
Fibonacci numbers in vegetables and fruit
Romanesque Brocolli/Cauliflower (or Romanesco) looks and tastes
like a cross between brocolli and cauliflower. Each floret is peaked
and is an identical but smaller version of the whole thing and this
makes the spirals easy to see.
© All rights reserved
Image Source >>
* * *
Every human has two hands, each one of these has five fingers,
each finger has three parts which are separated by two knuckles. All
of these numbers fit into the sequence. However keep in mind, this
could simply be coincidence.
To view more examples of Fibonacci numbers in Nature explore our
selection of related links>>.
Knowledge of the golden section, ratio and rectangle goes back to
the Greeks, who based their most famous work of art on them: the
Parthenon is full of golden rectangles. The Greek followers of the
mathematician and mystic Pythagoras even thought of the golden ratio
Later, Leonardo da Vinci painted Mona Lisa's face to
fit perfectly into a golden rectangle, and structured the rest of
the painting around similar rectangles.
Mozart divided a striking number of his sonatas into
two parts whose lengths reflect the golden ratio, though there is
much debate about whether he was conscious of this. In more modern
times, Hungarian composer Bela Bartok and French architect Le
Corbusier purposefully incorporated the golden ratio into their
Even today, the golden ratio is in human-made objects all around
us. Look at almost any Christian cross; the ratio of the vertical
part to the horizontal is the golden ratio. To find a golden
rectangle, you need to look no further than the credit cards in your
Despite these numerous appearances in works of art throughout the
ages, there is an ongoing debate among psychologists about whether
people really do perceive the golden shapes, particularly the golden
rectangle, as more beautiful than other shapes. In a 1995 article in
the journal Perception, professor Christopher Green,
University in Toronto, discusses several experiments over the years
that have shown no measurable preference for the golden rectangle,
but notes that several others have provided evidence suggesting such
a preference exists.
Regardless of the science, the golden ratio retains a mystique,
partly because excellent approximations of it turn up in many
unexpected places in nature. The spiral inside a nautilus shell is
remarkably close to the golden section, and the ratio of the lengths
of the thorax and abdomen in most bees is nearly the golden ratio.
Even a cross section of the most common form of human DNA fits
nicely into a golden decagon. The golden ratio and its relatives
also appear in many unexpected contexts in mathematics, and they
continue to spark interest in the mathematical community.
Dr. Stephen Marquardt, a former
plastic surgeon, has used the
golden section, that enigmatic number that has long stood for
beauty, and some of its relatives to make a mask that he
claims is the most beautiful shape a human face can have.
Mask of a perfect human face
Egyptian Queen Nefertiti (1400 B.C.)
An artist's impression of the face of Jesus
based on the Shroud of Turin and corrected
to match Dr. Stephen Marquardt's mask.
Click here for more detailed analysis.
"Averaged" (morphed) face of few
You can overlay the Repose
Frontal Mask (also called the RF Mask or Repose Expression –
Frontal View Mask) over a photograph of your own face to help you
to aid in evaluating your face for
face lift surgery, or
simply to see how much your face conforms to the measurements of the
Marquardt's Web site for more information on the beauty
Source of the above article (with
exception of few added photos):
The original problem that Fibonacci investigated, in the year
1202, was about how fast rabbits could breed in ideal circumstances.
"A pair of rabbits, one month old, is too young to reproduce.
Suppose that in their second month, and every month thereafter, they
produce a new pair. If each new pair of rabbits does the same, and
none of the rabbits dies, how many pairs of rabbits will there be at
the beginning of each month?"
- At the end of the first month, they mate, but there is still one
only 1 pair.
- At the end of the second month the female produces a new pair, so
now there are 2 pairs of rabbits in the field.
- At the end of the third month, the original female produces a
second pair, making 3 pairs in all in the field.
- At the end of the fourth month, the original female has produced
yet another new pair, the female born two months ago produces her
first pair also, making 5 pairs. (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html)
The number of pairs of rabbits in the field at the start of each
month is 1, 1, 2, 3, 5, 8, 13, 21, etc.
The Fibonacci Rectangles and Shell Spirals
can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1
next to each other. On top of both of these draw a square of size 2
Phi pendant gold - a Powerful Tool for Finding Harmony and Beauty
We can now draw a new square - touching both a unit square and
the latest square of side 2 - so having sides 3 units long; and then
another touching both the 2-square and the 3-square (which has sides
of 5 units). We can continue adding squares around the picture, each
new square having a side which is as long as the sum of the latest
two square's sides. This set of rectangles whose sides are two
successive Fibonacci numbers in length and which are composed of
squares with sides which are Fibonacci numbers, we will call the Fibonacci
The next diagram shows that we can draw a spiral by putting together
quarter circles, one in each new square. This is a spiral (the Fibonacci
Spiral). A similar curve to this occurs in nature as the shape
of a snail shell or some sea shells. Whereas the Fibonacci
Rectangles spiral increases in size by a factor of Phi (1.618..) in
a quarter of a turn (i.e. a point a further quarter of a turn
round the curve is 1.618... times as far from the centre, and this
applies to all points on the curve), the Nautilus spiral
curve takes a whole turn before points move a factor of
1.618... from the centre.
A slice through a Nautilus shell
These spiral shapes are called Equiangular
spirals. The links from these terms contain much more
information on these curves and pictures of computer-generated
Here is a curve which crosses the X-axis at the
The spiral part crosses at 1 2 5 13 etc on the positive axis, and
0 1 3 8 etc on the negative axis. The oscillatory part crosses at 0
1 1 2 3 5 8 13 etc on the positive axis. The curve is strangely
reminiscent of the shells of Nautilus and snails. This is not
surprising, as the curve tends to a logarithmic spiral as it
Nautilus shell (cut)
© All rights reserved.
Image source >>
Nautilus jewelry pendant gold - A Symbol of Nature’s Beauty
Proportion - Golden Ratio and Rule of Thirds
© R. Berdan 20/01/2004.
Published with permission of the author
Proportion refers the size relationship of visual elements to each other
and to the whole picture. One of the reasons proportion is often
considered important in composition is that viewers respond to it
emotionally. Proportion in art has been examined for hundreds of years,
long before photography was invented. One proportion that is often cited
as occurring frequently in design is the Golden mean or Golden ratio.
Golden Ratio: 1, 1, 2, 3, 5, 8, 13, 21, 34 etc.
Each succeeding number after 1 is equal to the sum of the two
preceding numbers. The Ratio formed 1:1.618 is called the golden
mean - the ratio of bc to ab is the same as ab to ac. If you
divide each smaller window again with the same ratio and joing
their corners you end up with a logarithmic spiral. This spiral
is a motif found frequently throughout nature in shells, horns
and flowers (and my Science & Art logo).
The Golden Mean or Phi occurs frequently in nature
and it may be that humans are genetically programmed to
recognize the ratio as being pleasing. Studies of top fashion
models revealed that their faces have an abundance of the 1.618
Many photographers and artists are aware of the rule of thirds, where a
picture is divided into three sections vertically and horizontally and
lines and points of intersection represent places to position important
visual elements. The golden ratio and its application are similar
although the golden ratio is not as well known and its' points of
intersection are closer together. Moving a horizon in a landscape to the
position of one third is often more effective than placing it in the
middle, but it could also be placed near the bottom one quarter or
sixth. There is nothing obligatory about applying the rule of thirds. In
placing visual elements for effective composition, one must assess many
factors including color, dominance, size and balance together with
proportion. Often a certain amount of imbalance or tension can make an
image more effective. This is where we come to the artists' intuition
and feelings about their subject. Each of us is unique and we should
strive to preserve those feelings and impressions about our chosen
subject that are different.
Rule of thirds grid applied to a landscape
Golden mean grid applied a simple composition
On analyzing some of my favorite photographs by laying down
grids (thirds or golden ratio in Adobe Photoshop) I find that some of my
images do indeed seem to correspond to the rule of thirds and to a
lesser extent the golden ratio, however many do not. I suspect an
analysis of other photographers' images would have similar results.
There are a few web sites and references to scientific studies that have
studied proportion in art and photography but I have not come across any
systematic studies that quantified their results- maybe I just need to
look harder (see link for more information about the use of the golden
In summary, proportion is an element of design you should always be
aware of but you must also realize that other design factors along with
your own unique sensitivity about the subject dictates where you should
place items in the viewfinder. Understanding proportion and various
elements of design are guidelines only and you should always follow your
instincts combined with your knowledge. Never be afraid to experiment
and try something drastically different, and learn from both your
successes and failures. Also try to be open minded about new ways of
taking pictures, new techniques, ideas - surround yourself with others
that share an open mind and enthusiasm and you will improve your
compositional skills quickly.
35 mm film has the dimensions 36 mm by 24 mm (3:2 ratio) - golden mean
ration of 1.6 to 1 Points of intersection are recommended as places to
position important elements in your picture.
Note: The above segment is part of the article COMPOSITION &
the ELEMENTS of VISUAL DESIGN by Robert Berdan ( http://www.scienceandart.org/
© R. Berdan 20/01/2004
Published with permission of the author.
The entire article can be found here (PDF):
Subject Related Links and Resources
Dance of the Planets
BLUEPRINTS OF THE COSMOS
- by Christine Sterne *****/*****
Patterns Photo CD 50 Royalty Free Images
Nautilus jewelry pendant gold - A Symbol of Nature’s Beauty
Bogomolny, A. "Golden Ratio in Geometry."
Hofstetter, K. "A Simple Construction of the
Golden Ratio." Forum Geom. 2, 65-66, 2002.
Hofstetter, K. "A 4-Step Construction of the
Golden Ratio." Forum Geom. 6, 179-180, 2006.
Olariu, A. "Golden Section and the Art of
Painting." 18 Aug 1999.
Sloane, N. J. A. Sequences
A114540 in "The On-Line Encyclopedia of Integer Sequences."
Weisstein, E. W. "Books about Golden Ratio."