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.

Surname 2

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.

Surname 3

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

Surname 4

(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

Surname 5

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.

Surname 6

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,

- Print.

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more