Golden Ratio

GOLDEN RATIO

DEFINITION

It is said that two quantities are in a golden ratio also known under names golden section, golden mean ^[1], medial section, golden cut ^[2], extreme and mean ratio ^[3] when the following, expressed in algebraic form, is true:

In other words if the whole is to the larger as the larger is to the smaller. The golden ratio is irrational mathematical constant. Its value is 1.6180339887... The golden ratio is symbolised by the Greek letter (phi) and also by the less known τ (tau).

This mathematical proportion is often recognized as 'aesthetically pleasing'. Because of that golden ratio found its use in many different fields like mathematics, architecture, geometry, science, biology, nature, art and design.

The actual discovery of the golden ratio can be lead back to the ancient Greeks and the Pythagoras. The greek mathematician Euclid of Alexandria (365BC - 300BC) mentioned the golden mean.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

Euclid of Alexandria(ca. 300BC)

CALCULATION

Method for calculation of golden ratio constant is beginning from the following algebraic formula:

From the formula we obtain that a is equal b/, so we can replace all occurences of a:

Devided by b formula changes to:

Now we need to multiply formula by and by moving content of the right side to the left side we get quadratic equation:

The only positive result to above equation is the irrational number we have already presented:

GEOMETRY AND MATHEMATICS

Fibonacci introduced to us, the Fibonacci-Numbers:

1,2,3,5,8,13,21,34,55,89,144,233,377,....

Every number of this alignment equals to 0,168 if you divide it with the one before it, and 1,618 when you divide it with the number after it. These are the relationships between the larger and smaller numbers in the golden ratio. Below you can see example for division of bigger by smaller number:

and by smaller devided by bigger number:

There are many shapes in which golden ratio were discovered. Examples of those you can see on the right side.

Pentagram	Golden rectangle

Image Illustrates the hidden golden ratio in the very special shape, pentagram.	A rectangle is a golden rectangle when the sides are in the 1:0,618 proportion.

ART & DESIGN

Since the beginning of ART & Design, the artist and designers used the golden ratio in order to manage and achieve beauty and balance in their creations. In the following table you can see few examples for all.

Mona-Liza	Aztek decorations	Credit cards

In Mona-Liza painting you can find many golden rectangles that together create golden spiral.	The space between the two heads is exacly Phi times the width of the heads.[Brooker]	if you measure a credit-card, the outcome would be a perfect golden rectangle.[Batterywholesaler]

Architecture

There is many examples of golden ratio even in architecture. Architect were inspired by this "sacred ratio" already in times of Pyramids. Even in Greek you can find golden rectangles, spirals, etc. In the table below you can find several examples.

Great Pyramid of Giza	Parthenon	Engineering Plaza

The Great Pyramid has proportions designed according to golden ratio.[Samuel Obara]	Even in proportions of Parthenon you can find golden spiral consisting of golden rectangles.[Samuel Obara]	The Designers of this new Plaza have chosen the Fibonacci series spiral, or also called the golden mean to design this new state of the art plaza.[Dr Knott, 1996-2005]

REFERENCES

[Kaphammel, 2000] Günther Kahammel, Der goldene Schnitt, 1. Auflage, Braunschweig, 2000

[Brooker] Carolyn Brooker, The Golden Ratio in the Arts, University of Bath, http://students.bath.ac.uk/ma1caab/art.html

[Dr Knott, 1996-2005] Dr Ron Knott, Fibonacci Numbers and The Golden Section in Art, Architecture and Music, University of Surrey, 1996-2005, http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html

[Batterywholesaler] Batterywholesaler, Credit Cards, http://www.batterywholesaler.co.uk/battery_images

[Samuel Obara] Samuel Obara, Golden Ratio in Art and Architecture, University of Georgia, http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html

[Livio, 2002] Mario Livio, The golden ratio and aesthetics, November 2002, http://plus.maths.org/issue22/features/golden/

[Samuel Obara]http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html

↑ Template:Cite book
↑ Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."
↑ Euclid, Elements, Book 6, Definition 3.

[livio-1] Template:Cite book

[2] Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."

[Elements_6.3-3] Euclid, Elements, Book 6, Definition 3.

[1]

[2]

[3]