Allah has appointed a measure for all things. (Surat
You will not find any flaw in the creation of
the All-Merciful. Look again-do you see any gaps? Then look again and
again. Your sight will return to you dazzled and exhausted! (Surat al-Mulk,
... If a pleasing or exceedingly balanced form is achieved in terms
of elements of application or function, then we may look for a function
of the Golden Number there ... The Golden Number is a product not of
mathematical imagination, but of a natural principle related to the
laws of equilibrium. (1)
What do the pyramids in Egypt, Leonardo do Vinci's portrait of the Mona
Lisa, sunflowers, the snail, the pine cone and your fingers all have in
The answer to this question lies hidden in a sequence of numbers discovered
by the Italian mathematician Fibonacci. The characteristic of these numbers,
known as the Fibonacci numbers, is that each one consists of the sum of
the two numbers before it. (2)
L. Pisano Fibonacci
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,
Fibonacci numbers have an interesting property. When you divide one number
in the sequence by the number before it, you obtain numbers very close
to one another. In fact, this number is fixed after the 13th in the series.
This number is known as the "golden ratio."
233 / 144 = 1.618
377 / 233 = 1.618
610 / 377 = 1.618
987 / 610 = 1.618
1597 / 987 = 1.618
2584 / 1597 = 1.618
THE HUMAN BODY AND THE GOLDEN RATIO
When conducting their researches or setting out their products, artists,
scientists and designers take the human body, the proportions of which
are set out according to the golden ratio, as their measure. Leonardo
da Vinci and Le Corbusier took the human body, proportioned according
to the golden ratio, as their measure when producing their designs. The
human body, proportioned according to the golden ratio, is taken as the
basis also in the Neufert, one of the most important reference books of
Leonardo da Vinci used the golden ratio in setting
out the proportions of the human body.
THE GOLDEN RATIO IN THE HUMAN BODY
The "ideal" proportional relations that are suggested as existing
among various parts of the average human body and that approximately meet
the golden ratio values can be set out in a general plan as follows: (3)
The M/m level in the table below is always equivalent to the golden ratio.
M/m = 1.618
The first example of the golden ratio in the average human body is that
when the distance between the navel and the foot is taken as 1 unit, the
height of a human being is equivalent to 1.618. Some other golden proportions
in the average human body are:
The distance between the finger tip and the elbow / distance between
the wrist and the elbow,
The distance between the shoulder line and the top of the head / head
The distance between the navel and the top of the head / the distance
between the shoulder line and the top of the head,
The distance between the navel and knee / distance between the knee and
the end of the foot.
The Human Hand
Lift your hand from the computer mouse and look at the shape of your
index finger. You will in all likelihood witness a golden proportion there.
Our fingers have three sections. The proportion of the first two to the
full length of the finger gives the golden ratio (with the exception of
the thumbs). You can also see that the proportion of the middle finger
to the little finger is also a golden ratio. (4)
You have two hands, and the fingers on them consist of three
sections. There are five fingers on each hand, and only eight
of these are articulated according to the golden number: 2, 3, 5, and
8 fit the Fibonacci numbers.
The Golden Ratio in the Human Face
There are several golden ratios in the human face. Do not pick up a ruler
and try to measure people's faces, however, because this refers to the
"ideal human face" determined by scientists and artists.
For example, the total width of the two front teeth in the upper jaw
over their height gives a golden ratio. The width of the first tooth from
the centre to the second tooth also yields a golden ratio. These are the
ideal proportions that a dentist may consider. Some other golden ratios
in the human face are:
Length of face / width of face,
Distance between the lips and where the eyebrows meet / length of nose,
Length of face / distance between tip of jaw and where the eyebrows meet,
Length of mouth / width of nose,
Width of nose / distance between nostrils,
Distance between pupils / distance between eyebrows.
Golden Proportion in the Lungs
In a study carried out between 1985 and 1987 (5),
the American physicist B. J. West and Dr. A. L. Goldberger revealed the
existence of the golden ratio in the structure of the lung. One feature
of the network of the bronchi that constitutes the lung is that
it is asymmetric. For example, the windpipe divides into two main bronchi,
one long (the left) and the other short (the right). This asymmetrical
division continues into the subsequent subdivisions of the bronchi. (6)
It was determined that in all these divisions the proportion of the short
bronchus to the long was always 1/1.618.
THE GOLDEN RECTANGLE AND THE DESIGN IN THE SPIRAL
A rectangle the proportion of whose sides is equal to the golden ratio
is known as a "golden rectangle." A rectangle whose sides are
1.618 and 1 units long is a golden rectangle. Let us assume a square drawn
along the length of the short side of this rectangle and draw a quarter
circle between two corners of the square. Then, let us draw a square and
a quarter circle on the remaining side and do this for all the remaining
rectangles in the main rectangle. When you do this you will end up with
The British aesthetician William Charlton explains the way that people
find the spiral pleasing and have been using it for thousands of years
stating that we find spirals pleasing because we are easily able to visually
follow them. (7)
The spirals based on the golden ratio contain the most incomparable designs
you can find in nature. The first examples we can give of this are the
spiral sequences on the sunflower and the pine cone. In addition to this,
an example of Almighty Allah's flawless creation and how He has created
everything with a measure, the growth process of many living things also
takes place in a logarithmic spiral form. The curves in the spiral are
always the same and the main form never changes no matter their size.
No other shape in mathematics possesses this property. (8)
The Design in Sea Shells
The flawless design in the nautilus shell contains
the golden ratio.
When investigating the shells of the living things classified as mollusks,
which live at the bottom of the sea, the form and the structure of the
internal and external surfaces of the shells attracted the scientists'
The internal surface is smooth, the outside one is fluted. The mollusk
body is inside shell and the internal surface of shells should be smooth.
The outside edges of the shell augment a rigidity of shells and, thus,
increase its strength. The shell forms astonish by their perfection
and profitability of means spent on its creation. The spiral's idea
in shells is expressed in the perfect geometrical form, in surprising
beautiful, "sharpened" design. (9)
The shells of most mollusks grow in a logarithmic spiral manner. There
can be no doubt, of course, that these animals are unaware of even the
simplest mathematical calculation, let alone logarithmic spirals. So how
is it that the creatures in question can know that this is the best way
for them to grow? How do these animals, that some scientists describe
as "primitive," know that this is the ideal form for them? It
is impossible for growth of this kind to take place in the absence of
a consciousness or intellect. That consciousness exists neither in mollusks
nor, despite what some scientists would claim, in nature itself. It is
totally irrational to seek to account for such a thing in terms of chance.
This design can only be the product of a superior intellect and knowledge,
and belongs to Almighty Allah, the Creator of all things:
"My Lord encompasses all things in His knowledge
so will you not pay heed?" (Surat al-An'am, 80)
Growth of this kind was described as "gnomic growth" by the
biologist Sir D'Arcy Thompson, an expert on the subject, who stated that
it was impossible to imagine a simpler system, during the growth of a
seashell, than which was based on widening and extension in line with
identical and unchanging proportions. As he pointed out, the shell constantly
grows, but its shape remains the same. (10)
One can see one of the best examples of this type of growth in a nautilus,
just a few centimetres in diameter. C. Morrison describes this growth
process, which is exceptionally difficult to plan even with human intelligence,
stating that along the nautilus shell, an internal spiral extends consisting
of a number of chambers with mother-of-pearl lined walls. As the animal
grows, it builds another chamber at the spiral shell mouth larger than
the one before it, and moves forward into this larger area by closing
the door behind it with a layer of mother-of-pearl. (11)
The scientific names of some other marine creatures with logarithmic
spirals containing the different growth ratios in their shells are:
Haliotis Parvus, Dolium Perdix, Murex, Fusus Antiquus, Scalari Pretiosa,
Ammonites, extinct sea animals that are today found only in fossil form,
too, had shells developing in logarithmic spiral form.
Growth in a spiral form in the animal world is not restricted to the
shells of mollusks. Animals such as antelopes, goats and rams complete
their horn development in spiral forms based on the golden ratio. (12)
The Golden Ratio in the Hearing and Balance Organ
The cochlea in the human inner ear serves to transmit sound vibrations.
This bony structure, filled with fluid, has a logarithmic spiral shape
with a fixed angle of ?=73°43´ containing the golden ratio.
Horns and Teeth That Grow in a Spiral Form
Examples of curves based on the logarithmic spiral can be seen in the
tusks of elephants and the now-extinct mammoth, lions' claws and parrots'
beaks. The eperia spider always weaves its webs in a logarithmic spiral.
Among the micro-organisms known as plankton, the bodies of globigerinae,
planorbis, vortex, terebra, turitellae and trochida are all constructed
THE GOLDEN RATIO IN THE MICRO WORLD
Geometrical shapes are by no means limited to triangles, squares, pentagons
or hexagons. These shapes can also come together in various ways and produce
new three-dimensional geometrical shapes. The cube and the pyramid are
the first examples that can be cited. In addition to these, however, there
are also such three-dimensional shapes as the tetrahedron (with regular
four faces), octahedron, dodecahedron and icosahedron, that we may never
encounter in our daily lives and whose names we may never even have heard
of. The dodecahedron consists of 12 pentagonal faces, and the icosahedron
of 20 triangles. Scientists have discovered that these shapes can all
mathematically turn into one another, and that this transformation takes
place with ratios linked to the golden ratio.
Three-dimensional forms that contain the golden ratio are very widespread
in micro-organisms. Many viruses have an icosahedron shape. The best known
of these is the Adeno virus. The protein sheath of the Adeno virus consists
of 252 protein subunits, all regularly set out. The 12 subunits in the
corners of the icosahedron are in the shape of pentagonal prisms. Rod-like
structures protrude from these corners.
first people to discover that viruses came in shapes containing the golden
ratio were Aaron Klug and Donald Caspar from Birkbeck College in London
in the 1950s. The first virus they established this in was the polio virus.
The Rhino 14 virus has the same shape as the polio virus.
Why is it that viruses have shapes based on the golden ratio, shapes
that it is hard for us even to visualise in our minds? A. Klug, who discovered
these shapes, explains:
My colleague Donald Caspar and I showed that the design of these viruses
could be explained in terms of a generalization of icosahedral symmetry
that allows identical units to be related to each other in a quasi-equivalent
way with a small measure of internal flexibility. We enumerated all
the possible designs, which have similarities to the geodesic domes
designed by the architect R. Buckminster Fuller. However, whereas Fuller's
domes have to be assembled following a fairly elaborate code, the design
of the virus shell allows it to build itself. (14)
Klug's description once again reveals a manifest truth. There is a sensitive
planning and intelligent design even in viruses, regarded by scientists
as "one of the simplest and smallest living things." (15)
This design is a great deal more successful than and superior to those
of Buckminster Fuller, one of the world's most eminent architects.
The dodecahedron and icosahedron also appear in the silica skeletons
of radiolarians, single-celled marine organisms.
Structures based on these two geometric shapes, like the regular dodecahedron
with feet-like structures protruding from each corner, and the various
formations on their surfaces make up the varying beautiful bodies of the
As an example of these organisms, less than a millimetre in size, we
may cite the icosahedron based Circigonia Icosahedra and the
Circorhegma Dodecahedra with dodecahedron skeleton. (17)
The Golden Ratio in DNA
The molecule in which all the physical features of living things are
stored, too, has been created in a form based on the golden ratio. The
DNA molecule, the very program of life, is based on the golden ratio.
DNA consists of two intertwined perpendicular helixes. The length of the
curve in each of these helixes is 34 angstroms and the width 21 angstroms.
(1 angstrom is one hundred millionth of a centimetre.) 21 and 34 are two
consecutive Fibonacci numbers.
The Golden Ratio in Snow Crystals
The golden ratio also manifests itself in crystal structures. Most of
these are in structures too minute to be seen with the naked eye. Yet
you can see the golden ratio in snow flakes. The various long and short
variations and protrusions that comprise the snow flake all yield the
golden ratio. (18)
THE GOLDEN RATIO IN SPACE
In the universe there are many spiral galaxies containing the golden
ratio in their structures.
The Golden Ratio in Physics
You encounter Fibonacci series and the golden ratio in fields that fall
under the sphere of physics. When a light is held over two contiguous
layers of glass, one part of that light passes through, one part is absorbed,
and the rest is reflected. What happens is a "multiple reflection."
The number of paths taken by the ray inside the glass before it emerges
again depends on the number of reflections it is subjected to. In conclusion,
when we determine the number of rays that re-emerge, we find that they
are compatible with the Fibonacci numbers.
The fact that a great many unconnected animate or inanimate structures
in nature are shaped according to a specific mathematical formula is one
of the clearest proofs that these have been specially designed. The golden
ratio is an aesthetic rule well known and applied by artists. Works of
art based on that ratio represent aesthetic perfection. Plants, galaxies,
micro-organisms, crystals and living things designed according to this
rule imitated by artists are all examples of Allah's superior artistry.
Allah reveals in the Qur'an that He has created all things with a measure.
Some of these verses read:
Allah has appointed a measure for all things.
(Surat at-Talaq, 3)
Everything has its measure with Him. (Surat ar-Ra'd,
1- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn
Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari,
2nd Edition, 1993, p. 155.
2- Guy Murchie, The Seven Mysteries of Life, First Mariner Boks, New York,
3- J. Cumming, Nucleus: Architecture and Building Construction, Longman,
4- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn Oran (The
Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd
Edition, 1993, p. 87.
5- A. L. Goldberger, et al., "Bronchial Asymmetry and Fibonacci Scaling."
Experientia, 41 : 1537, 1985.
6- E. R. Weibel, Morphometry of the Human Lung, Academic Press, 1963.
7- William Charlton, Aesthetics: An Introduction, Hutchinson University
Library, London, 1970.
8- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn Oran (The
Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd
Edition, 1993, p. 77.
9- "The 'Golden' spirals and 'pentagonal' symmetry in the alive Nature,"
online at: http://www.goldenmuseum.com/index_engl.html
10- D'Arcy Wentworth Thompson, On Growth and Form, C.U.P., Cambridge,
11- C. Morrison, Along The Track, Withcombe and Tombs, Melbourne.
12- "The 'Golden' spirals and 'pentagonal' symmetry in the alive
Nature," online at: http://www.goldenmuseum.com/index_engl.html
13- J. H. Mogle, et al., "The Stucture and Function of Viruses,"
Edward Arnold, London, 1978.
14- A. Klug, "Molecules on Grand Scale," New Scientist, 1561:46,
15- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn Oran
(The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari,
2nd Edition, 1993, p. 82.
16- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn Oran
(The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari,
2nd Edition, 1993, p. 85.
17- For bodies of radiolarians, see H. Weyl, Synnetry, Princeton, 1952.
18- Emre Becer, "Biçimsel Uyumun Matematiksel Kuralý
Olarak, Altýn Oran" (The Golden Ratio as a Mathematical Rule
of Formal Harmony), Bilim ve Teknik Dergisi (Journal of Science and Technology),
January 1991, p.16.
19- V.E. Hoggatt, Jr. and Bicknell-Johnson, Fibonacci Quartley, 17:118,