What is Phi, the Golden Ratio and the Fibonacci Series?


This thesis intends to analyze the usage of “ Aureate Numbers ” , since recorded clip began. It aims to compare the usage of Numberss, such as the Fibonacci Sequence, Phi and the Golden Ratio, within merchandise design today, and the execution of these mathematical sequences in the yesteryear. The ancient Italian Physicist and Mathematician, Galileo Galilei ( 1564 – 1642 ) , in his paper “ Opere Il Saggiatore ” , provinces:

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“ [ The Universe ] can non be read until we have learnt the linguistic communication and go familiar with the characters in which it is written. It is written in mathematical linguistic communication, and the letters are trigons, circles and other geometrical figures, without which means it is humanly impossible to grok a individual word. ” [ 1 ]

This emphasises the cognition of technology systems within mathematics, many 100s of old ages ago, nevertheless these subjects can be traced back much further, about 3000 old ages – to the clip of the Ancient Egyptians. The Babylonians and the Ancient Greeks are merely two civilizations, which can be seen to hold implemented Golden Numbers into the technology of their architectonics. [ 2 ]

This symmetricalness in nature has fascinated peoples of the universe for millenary. However, when one time these figure sequences were most frequently attributed to a causal consequence of a divinity, in today ‘s universe, a better apprehension of our existence has led most scientists to explicate this, repeat in nature, to facts and occurrences in the natural technology of our universe. [ 3 ]

Britishers in the 30th century BCE, the Ancient Egyptians, Hindus and the Ancient Greeks all studied nature and the universe about them every bit closely as they were able to. They observed that many natural constructions involve indistinguishable mathematical sequences and forms, such as the Fibonacci Spiral. They postulated that this repeat was in fact put in topographic point because of a reverent force or motive. Often, to emulate their several divinities ‘ supposed work, they copied these ‘natural technology designs ‘ into the art and architecture around them. [ 4 ]

Probably the most good known usage, for The Golden Ratio, is in the pictures of many creative persons from the Renaissance period. Many creative persons used proportions within their pictures, including Dali, Raphael, Vermeer and Da Vinci. [ 5 ];[ 6 ];It was because of the aesthetically pleasing nature of these creative persons ‘ plants that the Golden Ratio became known as the “ Divine Proportion ” .8.As can be seen in the construction of the Great Pyramid of Giza, the Golden Ratio offers immense advantages, when used in the architecture of edifices. [ 7 ]

In the modern universe, merchandise design is based on bring forthing functional and practical points which are both delighting to the oculus, ergonomic in nature and that are efficient to utilize. It can be seen that Golden Numbers are still used in the architecture of design, in a broad scope of modern merchandises. [ 8 ] This thesis will analyze how these Numberss and sequences are still used, in design, but non in the same context as they were in ancient times. Any important sacred intensions, to these technology systems, seem to hold diminished as the millenary and centuries have worn on. It will compare the usage of these Numberss, with mention to how religious importance has been steadily lessened, within design, as the old ages have moved from the first applied scientists ‘ times, to our modern universe, today.

Creusen and Shoormans, 2005, article in the Journal of Product Managment, discusses the cardinal rules of merchandise visual aspect, which consumers expect and demand:

“ Based on a literature reappraisal, six different functions of merchandise visual aspect for consumers are identified: ( 1 ) communicating of aesthetic, ( 2 ) symbolic, ( 3 ) functional, and ( 4 ) ergonomic information ; ( 5 ) attending pulling ; and ( 6 ) classification. A merchandise ‘s visual aspect can hold aesthetic and symbolic value for consumers, can pass on functional features and give a quality feeling ( functional value ) , and can pass on easiness of usage ( ergonomic value ) . In add-on, it can pull attending and can act upon the easiness of classification of the merchandise. ” [ 9 ]

Their qualitative survey, of consumers, showed that aesthetics and symbolism were the properties which caused most consequence, when consumers were taking merchandises.9.

Do I necessitate a “ shutting statement ” for the debut? ? ?.

MTPart1: What is Phi, the Golden Ratio and the Fibonacci Series?

It is of import to understand the significance and importance of these Numberss and sequences. It is besides important to hold an apprehension of what these Numberss and sequences are, and how they work, in mathematical and technology footings. This first subdivision intends to detect these facts and to explicate to the reader what they are and how they work.

The aureate ratio has possibly been viewed as a “ Godly Proportion ” because it is found about everyplace in the natural universe around us. Even the very composing of the human signifier can be shown to hold “ Aureate Proportions ” . Across nature Phi, the Fibonacci Sequence and the Golden Ratio can be found in the building of workss and animate beings, the layout of our planetal system and besides in the formation sequences of crystals. In the human ( unreal ) universe it is found in pictures, architecture and music. [ 10 ] The images and diagrams below, show illustrations of where these Numberss are found in the natural universe.

The Grecian mathematician, Euclid, who lived in Alexandria around 325bce to about 265bce, is widely recognised as the Father of Geometry. Despite it besides being recognised that the methods of geometry were used 1000s of old ages before Euclid was born, he was the first to closely analyze and enter the exactnesss of the topic. [ 11 ] .

The most important maxim that Euclid set down, in his Hagiographas about geometry, was that if a line is split in to two parts, “ a ” & amp ; “ B ” , and the ratio of the length of the amount of the measures to the length of the longer part is the same as the ratio of the lengths of parts a to B, this is a important and of import figure, which much later would go known as the “ Golden Ratio ” . [ 12 ] The diagrams below show this thought diagrammatically.

The aureate ratio, in mathematical footings is an irrational figure. This, as defined by The Encyclopaedia Britannica is:

“ … irrational figure [ s ] can be expressed as an infinitive denary enlargement with no regularly reiterating figure or group of figures. ” [ 13 ]

In this instance the irrational figure, in inquiry is 1.61803… It can besides be expressed as a quadratic equation:

Dunlap surveies the Golden Ratio and Fibonacci Numbers, in his book published in 1997. He instantly identifies that there are several definitions and names for these mathematical entities:

“ It has been called theaureate mean, theaureate subdivision, theaureate cut, theGodhead proportion, theFibonacci figureand themean of Phidias. ”[ 14 ]

Merely as this irrational figure has many pretenses and names, in mathematics it has different picks, for its ‘ symbol designator. Most normally, it is written as “ I• ” which is the Grecian small letter missive “ Phi ” . However, it can besides be found recorded as a lower instance Greek missive “ Tau ” ; “ ? ” , whilst the opposite value is given a appellation of “ ? ” , which is the uppercase Greek missive “ Phi ” .12

The proportions, set by the Golden Ratio, have been recognised as being used in the sculpture and architecture of Ancient Greece. The Parthenon in Athens explicitly uses the Golden Ratio in many facets of its design. Its interior decorator, Phidias, has now lent his name to the figure ‘s appellation, in mathematical and scientific expression, hence “ Phi ” . [ 15 ]

Despite the prevalence of cases, where Phi is found, it is non the most good known irrational figure. “ Pi ” or “ ? ” is about surely the most good known irrational figure. Pythagoras, yet another Grecian mathematician, who lived about DATE, besides examined geometry. Pythagorean Theorem describes the geometric belongingss of trigons and circles environing the principle that, in all right-angled trigons, the square of the amount of the hypotenuse ( the side opposite the right-angle ) is equal to the amount of the square of the other two sides. [ 16 ] The diagram and equation below, show how basic Pythagoras Theorem operates and is used in geometry.

The simplest equation, to happen the value of Phi is a quadratic expression, which is written as:

Finding the roots of this equation is carried out as follows:

ten =~ 1.618…orx=~ -.618…

The first of these replies is the figure, known as Phi ( 1.618… ) . The same consequence can be found through utilizing two different series operations, as so:

Phi =orPhi =

Taking the 2nd infinite series, that of infinite square roots, it can be demonstrated, in three easy stairss, how this equation correlates to the original quadratic equation:

Phi =

Measure 1: Square both sides…


Measure 2: Simplify the equation…

Measure 3: Reshuffle the equation, into the recognized quadratic agreement…

As shown these the operations are needed to bring forth the value of Phi. If one were to follow the series of fractions method, this would corroborate that Phi is the lone known figure, with in Mathematics which its reciprocal can be found, merely by deducting 1. [ 17 ]

As described earlier in this papers, the Golden Ratio is explicitly involved with geometry. Phi is found in many geometrical forms, this an alternate representation of this value to be demonstrated, alternatively of composing it as an irrational figure. As Euclid showed:

This can be displayed better, if lengths A and B are used as sides of a rectangle:

A rectangle which has these exact proportions is referred to as a “ Aureate Rectangle ” . When a perfect square ( a rectangle of precisely equal length sides ) , of side length “ 2 ” , were to be drawn ; and a perfect square of side length “ 1 ” was drawn following to it ; and so the losing borders were joined to organize a rectangle, as in the diagram below:

Then this would besides supply a “ Aureate Rectangle ” . If this procedure were to be repeated, so “ a series of twirling rectangles ” would look, as shown in the diagram below:

By taking the ratios of the length, of each rectangle, so it is possible to finally acquire to a figure value which straight correlates to Phi.

2/1 = 2 3/2= 1.5 5/3 = 1.666… 8/5 = 1.6 13/8 = 1.625 etc.

By plotting points, through the corners of the squares, on next sides, a Logarithmic Spiral is formed, as shown below:

A really similar form can be seen by utilizing isosceles trigons, which have base-angles of 72a?° , and bisecting one of said base-angles:

When this bisection of one of the 72a?° base-angles is repeated once more, and once more, “ a series of Whirling Triangles ” is created, as so:

Once once more, plotting the points of intersection, an Equiangular Spiral is formed:

Returning, for a minute, to the spiral formed, by the series of Golden Rectangles, it can be seen that the lengths of the sides of the rectangles, 1, 2, 3, 5, 8, 13, 21 etc. , organize a list of Numberss which is indistinguishable to the Fibonacci Series.17.The Fibonacci Sequence lends its name from the Italian mathematician, Leonardo of Pisa ( 1175 -1250ce ) .

Subsequently known as Leonardo Fibonacci, this now high mathematician, was the boy of Guglielmo Bonaccio, a imposts officer, from Pisa, Italy. Guglielmo sent for his boy, while he was in North Africa c.1192, and arranged for him:

“ … direction in calculational techniques, meaning for Leonardo to go a merchandiser. ” [ 18 ]

Leonardo went on to go avidly interested in Mathematicss and studied it abundantly, across Europe, the Middle-East and North-Africa. In 1200ce, he returned place to Pisa where he published a book, calledLiber Abaci(Book of Abacus ). This book was the first debut, to the Western concern universe, of the Hindu-Arabic Numberss, i.e. the Decimal Number System. The book shows how any figure can be constructed, utilizing the figures, 0 through to 9. In one subdivision of his book, Fibonacci discusses a theoretical job affecting coneies and their genteelness rates:

“ Suppose a newly-born brace of coneies, one male, one female, are put in a field. Rabbits are able to copulate at the age of one month so that at the terminal of its 2nd month a female can bring forth another brace of coneies. Suppose that our coneies ne’er die and that the female ever produces one new brace ( one male, one female ) every month from the 2nd month on. The mystifier that Fibonacci posed was… How many braces will at that place be in one twelvemonth? ” [ 19 ]

The reply, to this job, produces a series of Numberss now known as the “ Fibonacci Sequence ” .

No. of Rabbits













Calendar month













The regulation for bring forthing the Fibonacci Numberss can be expressed as:

Following through with this, formulaic device, in the 13th month one would hold a sum of 233 coneies.13.

Many centuries, before Fibonacci, Hippassus of Metapontum ( circa 450 bce ) , who was one of Pythagoras ‘ students, was able to turn out much to the alarm of Pythagoras, that the ratio of a diagonal of the regular Pentagon to the side length of the Pentagon could non be expressed as a fraction of whole numbers. The consequence was really found to be an irrational figure, which is tantamount to Phi ( ?¤ ) , one time once more this is the Golden Ratio.

In the diagram below, it can be seen that every diagonal, of a regular Pentagon, lies parallel to its opposite opposite number —the sides that it does non cross. As such, the sides of trigonsAEDand BTChold precisely parallel sides and so are regarded as similar to each other. mention Fabulous Fibonacci


Following this through means that:


This can be interpreted to state that:

Symbolically, we can compose this as:

( withvitamin Das the length of the diagonal andaas the length of the side ) .


The undermentioned equation is produced:

Once once more a quadratic equation can do, as so:

It can be seen thatis a positive root, which calculates to be Phi:

mention Fabulous Fibonacci

There is yet another premiss that has been identified, as being per se linked with Golden Numbers, and the similar. This construct was pointed out, by the Russian philosopher Shestakov, in DATE. Shestakov reiterated cognition of three cardinal rules of “ harmoniousness ” , which have been appreciated, and investigated since antediluvian times. His book, “ Harmony as an Aesthetic Class ” , he explains how the term “ harmoniousness ” has a various word picture and applies to perfectly everything in the existence, when depicting the manner that nature, infinite, beauty, and even morality, is arranged. He continues, by explicating that the ‘laws of aesthetic perceptual experience ‘ and the really regulations and dogmas of all right art design rely on “ Mathematical Harmony ” . The 2nd rule, in his aggregation of principles, is Aesthetic Harmony. He describes it, so:

“ In contrast to the mathematical harmoniousness, the aesthetic harmoniousness is non quantitative, but qualitative impression and expresses the internal nature of things. This aesthetic harmoniousness is connected with aesthetic exhilarations and appraisals. Most exactly this type of harmoniousness is shown at perceptual experience of beauty of Nature. ” referenceShestakov ( Harmony as an Aesthetic Category )

Shestakov, finishes his set with Artistic Harmony. This, as he explains, is associated with art, and how the rules of harmoniousness are realised in the field of art. referenceShestakov ( Harmony as an Aesthetic Category )

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