The Golden Ratio

The Golden Ratio is one of the most famous numbers in Mathematics. Plausibly, it is also
one of the most prominent numbers in arts, design, and architecture. The Golden Ratio refers to a
special value obtained by cutting a line into two so that the lengthier section of the line divided
by the smaller one is equal to the whole length of the line divided by the longer section. The
number is often symbolized using phi, after the 21st letter of the Greek alphabet. In its many
years of existence, the number has been discovered and rediscovered so many times as to have
different names used to refer to it. It is referred to as the Golden mean, the Golden section, or the
divine proportion (“The world’s most astonishing number” 124). Irrespective of what name one
assigns to it, there is no denying that the Phi relationship is the most universally binding of
mathematical relationships.
The history of Phi, particularly its application, dates back to ancient Greek times. Phidias,
a Greek sculptor, and mathematician, and the architect who designed the Parthenon is thought to
have applied Phi in his design. Plato considered the Golden ratio as the most comprehensive
mathematical relationship. The father of Geometry, Euclid, associated the Golden ratio to how a
pentagram is constructed. The most widely known application of Phi dates back to the
Renaissance period (Lehmann and Posamentier 49). In 1509, the mathematician Luca Pacioli
wrote a book that made reference to a number he called the “Divine Proportion.” To illustrate the
concept, he called upon the talents of his contemporary, Leonardo Da Vinci. Da Vinci later
referred to the same number as the Golden section or the sectio aurea.
The Golden ratio was used extensively as a tool for balance and beauty in Renaissance
paintings and sculptures. For example, Da Vinci made use of the Golden Ratio to define all of
the proportions in his infamous painting, the Last Supper. The dimensions of the table, walls and
the backgrounds all conform to the Golden Ratio. Also, the Golden ratio is found in Da Vinci’s
more famous paintings the Mona Lisa and the Vitruvian Man. The Golden ratio was also used by
other artist including Raphael, Michelangelo, Rembrandt, Seurat, and Salvador Dali. In
contemporary usage, the term “phi” was coined by Mark Barr, an American mathematician, in
the early 20 th century. Since then, Phi has found broad application in mathematics as well as the
related discipline of physics and even in art, architecture, and design.

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Figure 1 : Golden Ratio Illustration
As previously stated, the Golden Ratio arises from dividing a line segment so that the
ratio of the whole segment to the longer segment is equal to the ratio of the longer segment to the
smaller one. Euclid referred to it as division in extreme and mean ratio. Based on the figure
above, the Golden ration could be represented in equation form as:

. (1)
As with Pi (π), another important mathematical ratio, the digits go on and on, theoretically to
infinity. Alternatively, one could represent the Golden Ratio in terms of ϕ, as in Figure 2.

Figure 2: Golden Ratio in terms of ϕ
An alternative definition of ϕ can be derived from the figure above. Given a rectangle having
sides in the ratio 1:x, Phi is defined as the unique number x such that dividing the rectangle into a
new rectangle and a square as illustrated in Figure 2 above produces a new rectangle whose sides
have the same ratio 1:x. The side ratios for the two yellow rectangles shown in Figure 2 above
are similar. The successive points apportioning the Golden Rectangle into squares and rectangles
lie on a logarithmic spiral as seen in Figure 1 above. The resultant figure that maps squares onto
a logarithmic spiral is referred to as a whirling square.

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From the above explanation,

(2)

resulting in equation (3)

(3)

Figure 3: in Euclidian Geometry

From Euclidian Geometry, there is also an equivalent definition for , what Euclid defined in
terms of “extreme and mean ratios” on a line segment. From the segment in Figure 3, we can see
that

(4)

For the line segment AB above, we could plug in for the values to get

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(5)

If we got rid of the denominators by cross multiplying, we get equation 6

(6)

Which is the same formula for equation three above. From the results, we can surmise that is
an algebraic number of degree 2.) Using the equation 5, and using the positive value since the

figure cannot be equal to or less than 1 gives the value of ,

(6)

Phi also has some peculiarities unique to it in mathematics. For example, Phi is the only number
with a square that is greater than itself by one. The expression can be represented
mathematically as

(7)

Phi also happens to be the only number whose reciprocal is less than itself by one, expressed
mathematically as

(8)
The two qualities of Phi outlined above can be expressed algebraically as a+1=a² and a-1=1/a.
This can then be rearranged and expressed as a²-a -1=0, a quadratic equation, whose only
positive solution is ϕ = (1 + √5) /2 = 1.61803398874989484820…
As is common in mathematics, most concepts have a distinct correlation to other concepts
in the discipline. Around the year 1200 AD, the mathematician Leonardo Fibonacci discovered
the unique properties of the sequence that was to carry his name. The Fibonacci sequence has a
close relation to the Golden Ratio. On taking any two succeeding Fibonacci numbers, one finds
that their ratio is almost equal to the Golden ratio (Dunlap and Dunlap 142). As the numbers in

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the sequence increase, the value gradually transforms to 1. 61803398874989484820…., which is
Phi. Taking the ratio of 3 and 5, for example, would give the ratio of 1.666. The ratio for 13 to 21
is 1.625 while that of 144 to 233 is 1.618, almost the equivalent of Phi. The Fibonacci sequence
numbers can also be mapped directly onto the Golden rectangle (Dunlap and Dunlap 153). The
Golden rectangle also relates to the Golden spiral, which is produced by taking adjacent squares
of Fibonacci dimensions to form the whirling square.
The Golden Ration is expansively present in nature. From flower petals, to seed heads in
the sunflower, pinecones, and even Nautilus shells. Even more intriguing yet is the extensive
appearance of Phi all over the human form, be it in the face, fingers, body, and even our DNA.
Subsequently, the ratio has had a profound impact on man’s perception of human beauty.
Evidence exists to support the fact that we perceive beauty in both sexes depending on how
closely the proportions of their facial and body dimensions come to Phi (Lehmann and
Posamentier 53). It would seem that Phi exists in our consciousness as a guide to beauty. For this
reason, it has found wide application in art, architecture, design, and engineering (Walser 8).
Architects, artists, engineers, and designers have long made use of Phi to capture the beauty and
harmony of nature in their creations.
Two of the greatest marvels of ancient architecture, the Great Pyramid of Egypt and the
Pantheon appear, to embody the Golden Ratio (“The world’s most astonishing number” 42). In
contemporary architecture, Notre Dame in Paris, the CN Tower in Toronto, and the UN
Secretariat building in New York all make use of the ratio. Even the Mona Lisa and the treasured
Stradivarius violins built around 1700 show Phi relationships (“The Extraordinary Number of
Nature” 3). It is a well-regarded fact that Leonardo Da Vinci, along with other artists including
Sandro Botticelli, Raphael, and Georges Seurat used Phi, then known as “The Divine
Proportion” (“The Extraordinary Number of Nature” 3). Phi has also had useful application in
design, featuring in company logos. In the 1970s, the use of Phi permitted surfaces to be lined in
five-fold symmetry in Penrose Tiles (“The world’s most astonishing number” 201). So prevalent
is the number that it has used in high fashion clothing design, and is the basis for ‘The Fashion
Code’, a style guide to women’s dress.
In conclusion, Phi is a very prevalent and useful mathematical concept. The practicality
of the idea has been discovered by successful generations throughout history from ancient
Greeks to Renaissance artists and even contemporary architects. The quality and usefulness of
the ratio are evidenced in the enduring monuments and classical artworks, some of which date
back to the beginning of time. Not only is it useful in the field of mathematics, but it has also
found wide application in the areas of physics, engineering, arts, design, and architecture among
other disciplines. The Golden ratio is indeed golden. It is no wonder that the Golden ratio has
come to be considered as the most universally binding of mathematical relationships.

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Works Cited

Dunlap, Richard A., and R. A. Dunlap. The golden ratio and Fibonacci numbers. Hackensack,
NJ: World Scientific, 1997.
Lehmann, Ingmar and Alfred S. Posamentier. “The Glorious Golden Ratio”. Choice Reviews
Online 49.10 (2012): 49-5723-49-5723. Web.
Livio, Mario. The Golden Ratio: The Story of Phi, the Extra Ordinary [ie Extraordinary]
Number of Nature, Art and Beauty. Review, 2003.
Livio, Mario. The golden ratio: The story of phi, the world’s most astonishing number. Broadway
Books, 2008.
Walser, Hans. The Golden Section. [Washington, D.C.]: Mathematical Association of America,

2001. Print.