Golden Rectangle and Golden Ratios

7/30/2019 Golden Rectangle and Golden Ratios

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Introduction

Many 20th centuryartistsandarchitectshave proportioned their works to approximate the golden ratio

especially in the form of thegolden rectangle, in which the ratio of the longer side to the shorter is thegolden ratiobelieving this proportion to beaestheticallypleasingMathematicianssince Euclid have

studied the properties of the golden ratio, including its appearance in the dimensions of aregular

pentagonand in agolden rectangle, which can be cut into a square and a smaller rectangle with the

sameaspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as

well as man-made systems such asfinancial markets, in some cases based on dubious fits to data.[10]
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Golden Ratio

Line segments in the golden ratio

Inmathematicsand thearts, two quantities are in the golden ratio if theratioof thesumof the quantities

to the larger quantity isequal tothe ratio of the larger quantity to the smaller one. The figure on the right

illustrates the geometric relationship. Expressed algebraically:

where the Greek letterphi( ) represents the golden ratio. Its value is:

[1]

The golden ratio is also called the golden section (Latin: sectio aurea) orgolden mean.[2][3][4]

Other

names include extreme and mean ratio,[5]

medial section, divine proportion, divine

section (Latin: sectio divina), golden proportion, golden cut,[6]

and golden number.

Calculation[edit]

Two quantities a and b are said to be in the golden ratio if:

One method for finding the value of is to start with the left fraction. Through simplifying the fraction and

substituting in b/a = 1/,

it is shown that

Multiplying by gives

which can be rearranged to
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Using thequadratic formula, two solutions are obtained:

and

Because is the ratio between positive quantities is necessarily positive:

.

ApplicationsAesthetics

Architecture

Painting

Book design

Industrial design

Music
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Golden Rectangle

Agolden rectanglewith longer side aand shorter side b, when placed adjacent to a square with sides of length a, will

produce asimilargolden rectangle with longer side a + band shorter side a. This illustrates the

relationship .

A golden rectangle is one whose side lengths are in thegolden ratio, , which is (the

Greek letterphi), where is approximately 1.618.

A distinctive feature of this shape is that when asquaresection is removed, the remainder is another

goldenrectangle; that is, with the sameaspect ratioas the first. Square removal can be repeatedinfinitely, in which case corresponding corners of the squares form an infinite sequence of points on

thegolden spiral, the uniquelogarithmic spiralwith this property.

According to astrophysicist and mathematics popularizerMario Livio, since the publication ofLuca

Pacioli's Divina Proportione in 1509,[1]

when "with Pacioli's book, the Golden Ratio started to become

available to artists in theoretical treatises that were not overly mathematical, that they could actually

use,"[2]

many artists and architects have been fascinated by the presumption that the golden rectangle is

considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works

predating Pacioli's publication.[3]

Construction
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A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio.

A golden rectangle can beconstructed with only straightedge and compassby this technique:

1. Construct a simple square

2. Draw a line from the midpoint of one side of the square to an opposite corner

3. Use that line as the radius to draw an arc that defines the height of the rectangle

4. Complete the golden rectangle.

Applications [edit]

Le Corbusier's 1927Villa SteininGarchesfeatures a rectangular ground plan, elevation, and inner

structure that closely approximate golden rectangles.[4]

Theflag of Togowas designed to approximate a golden rectangle.[5]
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Conclusion

Supposedly, Pythagoras discovered this ratio. And the ancient Greeks incorporated it into their

art and architecture. Apparently, many ancient buildings (including the Parthenon) use golden

rectangles. It was thought to be the most pleasing of all rectangles. It was not too thick, not too

thin, but just right (Baby Bear rectangles).

Because of this, sheets of paper and blank canvases are often somewhat close to being golden

rectangles. 8.5x11 is not particularly close to a golden rectangle, by the way.

The golden ratio is seen in some surprising areas of mathematics. The ratio of consecutive

Fibonacci numbers (1, 1, 2, 3, 5, 8, 13 . . ., each number being the sum of the previous two

numbers) approaches the golden ratio, as the sequence gets infinitely long. The sequence is

sometimes defined as starting at 0, 1, 1, 2, 3 . . . Zero is the zeroth element of the sequence.

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Golden Rectangle and Golden Ratios