Allah has
appointed a measure for all things. (Surat at-Talaq, 3)
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, 3-4)
... 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 common?
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 Fibonacci numbers
0, 1, 1, 2, 3, 5,
8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,
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."
GOLDEN RATIO = 1.618
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
modern-day architects.
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 length,
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 a spiral.
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' attention:
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, Solarium Trochleare. 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 ?=7343 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 on
spirals.
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.
The 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 radiolarians.
(16)
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,
8)
1- Mehmet Suat Bergil, Doada/Bilimde/Sanatta, Altn
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, pp. 58-59.
3- J. Cumming, Nucleus:
Architecture and Building Construction, Longman, 1985.
4- Mehmet Suat
Bergil, Doada/Bilimde/Sanatta, Altn 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, Doada/Bilimde/Sanatta, Altn 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, 1961.
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, 1987.
15-
Mehmet Suat Bergil, Doada/Bilimde/Sanatta, Altn Oran (The Golden Ratio in
Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 82.
16- Mehmet Suat Bergil, Doada/Bilimde/Sanatta, Altn 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, "Biimsel Uyumun Matematiksel Kural Olarak, Altn
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,
1979.
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