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Running head: GOLDEN RATIO 1
Golden Ratio
Student’s Name
Institution
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GOLDEN RATIO 2
Golden Ratio
CHAPTER 1: Introduction
A mathematical theorem is defined as a statement that has been proven by valid and
logical inference within a formula (Chu-Carroll, 2007). Inferences used in proving a theorem,
must be from important axioms of the same theorem. Golden ration is also referred to as divine
proportion. It is not exactly known when golden ratio was discovered but it is believed that it
was discovered and rediscovered throughout history. The first application of golden ratio is not
well known since it was being rediscovered constantly and that different names were used to
refer the ratio. Golden section, golden mean, golden number, divine section, and golden section
are the synonyms used to refer to golden ratio in by different scholars from different regions.
Golden ratio is written as the Greek letter phi whose mathematical symbol is the value of phi
is approximated to be 1.618. The golden ration is a complex equation that any two numbers can
be plugged into. The ratio is in the form:
= = phi.
The answer to the golden ratio is always yield a unique irrational number; therefore
mathematicians believe that this ratio is the most irrational number in mathematics. Mostly, the
numbers that result from calculation of phi are too complex therefore it equation are commonly
graphed. However, the process of graphing equation is difficult. The easiest way to construct
golden ratio is to construct a “1” square and a point is placed halfway through the side of the
square. Then, a line is drawn to the opposite corner from the point and turns it so the line is on
the same side as the dot was previously placed on. Lines are extended to create a golden
rectangle that contains the golden ration within it. Simply, the formula used to create the circle
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GOLDEN RATIO 3
can be written as ½ + /5 /2 = 1+ /5 /2 = 1.618 which proves that it was cosnstructed by the
golden ratio.The very major aplications of golden ratio started in greece and egypt. Biers
&William (1996) found that the most mathematicians such as Phidias began to study golden
ratio with the aim of discovering its importance to real life applications. Phidias used this ratio in
various ways which included the well known sculptures for the Parthenon. Phidias succeeded in
his artistic work inspiring many artists to study more of the golden ratio application.
Application of the golden ratio attracted the great philosopher plato after the
construction of sculptures for the Parthenon by Phidias. Michael (2004), Plato stated that the
golden mean is the most binding of all mathematical relationships and the key to the physics of
cosmos. He stated that a philosopher by the name Euclid discovered that golden ratio cab be used
to construct a pentagram. The basics and the characteristics off golden ratio are still in use even
today.
The experts have been searching for the application of golden ratio since the Greek’s
rediscovery in the world around us. Most things have been found related to golden ratio as the
experts have searched more and more. Golden ratio is found in everything, from architecture to
art, music to plants and can also be found in humans. Golden ratio is everywhere whether we
tend to realise it when looking for beauty or subconsciously.
Golden ratio is used to design the world’s most famous architectural structures. The great
Pyramid of Giza for instance, when you set half of the base equal to one, the slant height turns
out to equal to Phi. The Renaissance artist, used golden ratio through the construction or Notre
Dame. Golden ratio can also be found in modern architecture; for instance the United Nations
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GOLDEN RATIO 4
building, when you compare the width of the building with the height of every tenth floor is a
golden ratio.
Golden ratio has a great impact on art since it helps to beautify artistic work if properly
implemented. James (2007), found that Leonardo Da Vinci used golden ratio often on his famous
works. Mona Lisa turned out to be a perfect golden rectangle which was beautiful; this was
realized through the calculating ratio of the width of her forehead compared to the length from
the top of her head to her chin. Last Supper was his masterpiece to which contained golden ratio
in several places, this was seen in both the ceiling and the position where the people sit. The
entire human body is proportioned according to golden ratio as it is illustrated by Virtruvian Man
work of art.
The golden ratio and Fibonacci numbers are related according to nature. A growing plant
process follows the Fibonacci numbers, from the first shoot, to two shoots, to thirteen and so on.
The spacing of the leaves around each branch or stalk spiral it’s all golden ratio. In flowers
Fibonacci numbers are demonstrated through the sepal, outer petals, and inner petals on the back,
as well as stamen and carpel’s on the front. Everything that entails a flower follows the
Fibonacci Numbers including the petal count.
Golden ration can also be used to create a human mask that embodies phi in all
dimensions. This mask is claimed to be the most beautiful shape a human face can ever have for
it uses decagons and pentagons as its function. The origin of facial attractiveness can be argued
that, it can be traced back into mathematical terms with the golden ratio. The human smile can
also be found to have the golden ratio. This exists in the height to width of the center two teeth,
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GOLDEN RATIO 5
the ratio for the two center teeth to those next to them is phi, and the ratio of the width of the
smile to the third tooth from the center is also phi.
Golden ratio can be found everywhere, be it in ancient structures from thousands of years
ago to the new technology that we have today. Golden ratio has been in existence and it is
application to the day-to-day life will never end.
CHAPTER 2
The human face takes after the golden ration which sometimes is referred to the golden
section or divine proportion. In order to understand the shape and design of the face of a
beautiful lady, golden ratios are applied repeatedly. According to the golden ratios, if the
proportions of the face distances is close or equal to the value of phi =1.61 the lady is beautiful.
In this paper, both vertical and horizontal proportions of human face in regard to the golden ratio
will be discussed in detail.
VERTICAL PROPORTIONS
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