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Abiram Devnathan, 14­MT­209
Department of Mathematics
Loyola College,Chennai
The study is to emphasis the importance of ‘Golden Ratio’,an impeccable mathematical tool
which exist in nature.The research work gradually shifts from the mathematical explanation of
golden ratio to its application in designs.The aspects explored are: the history of golden
ratio;golden ratio and its usage in both ancient and modern architecture;golden ratio in internet
The golden ratio also is called the golden mean or golden section. Mathematicians since ​Euclid
have studied the properties of the golden ratio, including its appearance in the dimensions of a
regular pentagon​ and in a golden rectangle, which may be cut into a square and a smaller
rectangle with the same ​aspect ratio​. Miraculously, it also exist in the appearance of galaxies.The
golden ratio has been used to analyze the proportions of natural objects as well as man­made
systems such as ​financial markets​, in some cases based on dubious fits to data. Today almost all
web based technologies use the concept of golden ratio. Since it majorly involves geometrical
configurations, fields like architecture,web designing, modern art,rely on this natural
mathematical tool. All modern day designings use golden ratio but developed through the
modern tech tools. People use this knowingly or unknowingly in their design works.
Interestingly,it is highly pleasing and stable natural appearance which can be perceived by
human eyes. Since its natural existence is found, it was also termed as ‘Divine Ratio’ or ‘Divine
Proportion’. How is it possible? Right from the design of galaxies to the design pattern in the
stable seeds of sunflower, human beings are able to find the presence of this spectacular
mathematics. This ratio is found prevalently in three forms.They are: geometrical curve,
algebraical number and as polygonal structures. In later part of the research, mathematicals
details are elucidated.
Review of Literature
Golden ratio is not mathematics alone. It is beyond that. Art and mathematics carries out a
synergic relationship with each other. Experts and maverick geniuses across the world had and
have realized it. Each of them,be it: Newton, Einstein or Leonardo Da Vinci ,they have recorded
this realization through a quote on what do mathematics and art mean for them. Johannes Kepler,
the western astronomer who strived to understand the solar system,said the following.​“Geometry
has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into
extreme and mean ratio”[1].This synergic reference is found in Chapter 13 of Bhagvat Gita,
which says: “Without mathematics there is no art”.​ People who are gifted with creativity
excelled in the field of art though their profession was some other field. Richard Phillip
Feynman, popularly named as the Physicist of 20th century was a artist. People belonging to this
intellectual plane were able to visualise the math behind all the art works. Obviously, golden
ratio is not an exception.In Living Philosophies, a collection of personal philosophies of famous
people published in 1931, Albert Einstein has said :​“The most beautiful thing we can experience
is the mysterious. It is the source of all true art and science”[2]. It was Einstein’s direct reference
to the natural existence of phenomenal concept such as golden ratio and Theorem of Pythagoras.
Other contemporary professors and authors who have worked on golden ratio have also made
points which support this research work. Md. Akhtaruzzaman says “The Golden Proportion is
considered as the most pleasing to human visual sensation and not limited to aesthetic beauty but
also be found its existence in natural world through the body proportions of living beings, the
growth patterns of many plants, insects and also in the model of enigmatic universe”[3].In an
article​ ​Z. Kazlacheva and J. Ilieva,the say that ​“​The proportions of the Golden ratio and
Fibonacci sequence associate harmony and beauty and by this reason they are used in design”[4].
Mathematics behind golden ratio
Golden ratio is simply but a mathematical number which has its value as 1.6180339887.. . It is
also mathematically expressed as ​ ​Φ(Phi). ​It is an irrational number. In mathematics the number
system is classified as ‘Real numbers’ and ‘Complex numbers’. Under real numbers, it is further
classified as ‘Rational numbers’ and ‘Irrational numbers’. The rational numbers are the numbers
which terminates or better described as the one which has finite numbers after its decimal point.
On contrary, irrational numbers are the one which never terminates or the one which never ends.
The golden ratio is an irrational number whose value never ends. Other prominent ratios are
π(Pie) whose value is 3.14159..and e (Euler’s number) whose value is ​2.7182.. . Same as the
golden ratio, both pie and Euler’s number are irrational number which do not terminate. But
what does it mean by the term ‘ratio’? Firstly, ratio is a relationship between two numbers
indicating how many times the first number contains or contained in the second. Example are:
2/7, 5/2,etc.
Now what does it mean by saying ‘Golden Ratio’? Two quantities are said to be in golden ratio if
their ratio is same as the ratio of their sum to the larger of the two quantities.It is mathematically
expressed as :
In the above expression, it is required to note that the value of a is greater than value of b which
in turn is greater than o(a>b>0). The value attained for the above expression is
1.618(approximately) and this ratio is called as golden ratio. Golden ratio is also expressed
algebraically as:
From the above equations we write golden ratio(Phi) as :
Therefore, we can also understand from these equations that this ratio is irrational.
The above system continues to prove that this sequence is never ending[5]. In the year 1200 AD,
Leonardo Fibonacci found an interesting mathematical sequence.It is called as Fibonacci
sequence.To understand its relationship with golden ratio, consider the following sequence.
In the above sequence, any number(except the first two numbers) is expressed as the sum of
previous two numbers. Now consider any two successive numbers from the third number in the
series. ​When we take any two successive (one after the other) Fibonacci Numbers, their ratio is
very close to the Golden Ratio[6]:
Golden ratio in ancient architecture
Before going on to ancient architecture, an understanding on golden rectangle,golden triangle
and golden curve is necessary.Consider the following figures.
The figure.1 represents the golden rectangle in which it re­emphasizes the Equation.1
above.Secondly, the figure.2 represents the golden spiral. And thirdly, the figure.3 represents the
golden triangle along with the golden curve.The Golden Rectangle can be used to create a spiral,
the Golden Spiral. Starting with one Golden Rectangle, a second Golden Rectangle can be
attached to the first using the longest side of the rectangle, side A as the shortest side B of the
next rectangle. To this end the second rectangle is constructed 90 degrees perpendicular to the
first rectangle. If this process is continued, called the spiraling of the Golden Rectangle, a curved
line can be drawn through the corners of the rectangles creating the Golden Mean spiral. The
spiraling of the Golden Mean spiral continues indefinitely in inward and outward directions, it's
getting smaller and smaller spiraling inwards and getting bigger and bigger spiraling
outwards[7].It also shares its connection with the fibonacci sequence,which is elucidated in the
table above.With such a mathematical background, golden ratio was used in ancient and modern
architectural designs. The construction of ‘The Parthenon’ is a suitable example.
The Parthenon is a former temple in Acropolis,Greece. This temple is dedicated to goddess
Athena.The construction of the temple embarked in 447 BC. Similar to many other temples in
Greece, the design of ‘The Parthenon’ is also in such a way,to be seen only from outside.It is
also called as facade architecture. Naturally, golden ratio or the golden rectangle concept is used
which is seen in the figure.4. It is very appealing and pleasing to our human eyes.Some experts
also say that this construction had unknowingly used the concept of golden ratio which again
highlights that it is the most stable architectural configuration. Similarly,t​he Great Pyramid of
Giza's​ dimensions are also based on the Golden Ratio. If we take a cross section of the Great
Pyramid, we get a right triangle, the so­called Egyptian Triangle. The ratio of the slant
meta­height of the pyramid (hypotenuse of the triangle) to the distance from ground center (half
the base dimension) is 1.61804 ... which differs from phi by only one unit in the fifth decimal
place. If we let the base dimension be 2 units, then the sides of the right triangle are in the
proportion 1:sqrt(phi):phi and the pyramid has a meta­height of sqrt(phi)[7].
Golden ratio in modern architecture
Modern architects take advantage of golden ratio. They find that it is highly appealing to our
eyes.More importantly the mathematics behind it provides a proper balance for the entire
constructional system. As stated above, the golden ratio gives various options for construction. It
can be used as a rectangle or a polygonal construction. This is very prevalent in traditional and
cultural countries. But modern buildings in Japan,Singapore and some European nations are
spiral architectures, which have used the golden spiral. India’s richest man, Mukesh Ambani’s
official modern residence in the city of Mumbai was constructed with the very same idea of
golden ratio that was used in the ancient architectures.
Golden ratio in web layouts
According to experts in the field of web designing, they say that websites are not appealing
because of plethora of colours used. Rather, it is because of the grid layout. This usage is an
explicit application of divine proportion or golden rectangle.Normally, the websites are
appealing when they use the below standard layout which is made of divine rectangle.
We perceive visual appeal based on ratio. For thousands of years artists, designers, architects,
etc. have either intentionally or unintentionally used a common ratio in their work that is
aesthetically pleasing.
Using the golden ratio is very simple. If we say we want to find the width of our main content
and Sidebar columns. We will take the total width of our content area (we will use 900px for this
example) and divide that by 1.62. As shown in the example above we divide 900px by 1.62 and
get 555.55px. We don't need to be exact so we will round it off to 555px. Now we know our
main content element will be 555px wide[8]. Naturally web designers are more relying upon the
golden ratio for stable layout configuration. Websites are developed with the help of tools such
as HTML5 and Cascading Style Sheets(CSS). With these tools, web developers will design
values that are satisfying the divine ratio properties,similar to the example shown above.
Existence of divine ratio in nature
The concept of golden ratio is already there in the nature. For this reason, the golden ratio is also
called as divine ratio which is a direct reference to the existence of god. The most pertinent
example is found in Sunflower. The arrangement of the seeds of the Sunflower is a spiral curve
or the golden curve. Scientists call this arrangement as a physically stable one and visually
appealing one. The seeds are intact to each other. This is to prevent the seeds detaching from the
flower by experiencing only an iota current of air. If we magnify this visionary million times, we
will find that, even our galaxies in this universe are spiral in shape. It is not a normal spiral. As
mentioned above, it is the divine spiral. Similarly, it is also found in the face of human beings,
ears of certain animals and also in plethora of flora and fauna.
Using Fibonacci numbers, the Golden Ratio becomes a golden spiral, that plays a vital role
everywhere in the nature such as in shells, pine cones, the arrangement of seeds in a sunflower
head and even galaxies. Adolf Zeising, a mathematician and philosopher, while studying the
natural world, saw that the Golden Ratio is operating as a universal law. Midhat J.Gazale says
that until Euclid the golden ratio and its mathematical properties were not studied. In the “
Misconceptions about the Golden Ratio”, Dr. George Markowsky also discussed about some
misconceptions of the properties and existence golden ratio[9] in various structures and design.
Basically the Golden Ratio should not be considered as a convention to all circumstances like a
law of nature but it needs deeper study and analysis to establish the relation with the ratio as it is
a curiosity of researchers to fulfil the demand of this field of research. Basically this study
represents a qualitative view on Golden Proportion from ancient time to the modern days. The
study also represents the mystery of various geometrical patterns and various dynamic rectangles
which are found in nature[10]. The mathematical explanation of the equation of Phi based on the
classical approach is also elucidated in this paper. Mainly the paper explains how this world is
full of designs made of divine ratio;both natural and man­made.
[1] Kepler, Johannes, (Ubi materia, ibi geometria.) J. Koenderink Solid Shape, Cambridge
Mass.: MIT Press, 1990,February 2016­02­25
[2]Simon and Schuster.Living Philosophies,Vol 13 ,What I believe,1931. Pg 193,No and Pg
194,No.February 2016­02­25
[3]M. Akhtaruzzaman and A. A. Shafie.Geometrical Substantiation of ​ Phi ​, the Golden Ratio
andthe Baroque of Nature, Architecture,Design and
_Ratio_and_the_Baroque_of_Nature_Architecture_Design_and_Engineering​. February
[4]Z. Kazlacheva and J. Ilieva.The Golden and Fibonacci Geometry in Fashion and Textile
[5] Golden Ratio.​https://en.wikipedia.org/wiki/Golden_ratio​. February 2016­02­25
[6]Nature, The Golden Ratio,and Fibonacci too…
http://www.mathsisfun.com/numbers/nature­golden­ratio­fibonacci.html​. March 2016­03­03
[7]The Golden Ratio. h​ ttp://www.tokenrock.com/explain­golden­ratio­177.html​. March
[8] Remick,Jarel. ​The Golden Ratio in Web Design.
http://code.tutsplus.com/tutorials/the­golden­ratio­in­web­design­­net­2272​.March 2016­03­11
[9]G. Markowsky. Misconceptions about the Golden Ratio. The College Mathematics Journal,
Vol. 23, No. March 2016­03­15
[10] T
​ he Golden Ratio: Phi, 1.618.​http://www.goldennumber.net/​. March 2016­03­15