Golden Ratio

This article is about the mathematical ratio. For the Aristotelian concept of "golden mean" see Nicomachean Ethics.

The golden ratio is an irrational number, approximately 1.618, that possesses many interesting properties. Shapes defined by the golden ratio have long been considered aesthetically pleasing in Western cultures, reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios. The golden ratio is still used frequently in art and design. The golden ratio is also referred to as the golden mean, golden section, golden number or divine proportion.

The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry. The golden ratio may have been understood and used by the Egyptians. The discovery of irrational numbers, numbers that cannot be represented as an exact ratio of two integers, is usually attributed to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Euclid spoke of the "golden mean" this way, "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".

The golden ratio is typically symbolized by the Greek letter φ (phi), with τ (tau) being less common.

<math>\varphi = \frac{\sqrt{5} + 1}{2} \approx 1.618 033 988 749 894 848 204 586 834 366 \ <math>

Origin of name

The name "golden ratio" appears in the form sectio aurea, "golden section", by Leonardo da Vinci. The American mathematician Mark Barr first conceived of the use of the symbol φ to represent the golden ratio, taking it from the first letter in the name of the Greek sculptor Phidias, who was long believed to have used the golden ratio in his designs.

Definition

Two quantities are said to be in the golden ratio, if "the whole is to the larger as the larger is to the smaller", i.e. if

<math>\frac{a+b}{a} = \frac{a}{b}.<math>

Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference:

<math>\frac{a}{b} = \frac{b}{a-b}.<math>

After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are seen to be equivalent to

<math>\left(\frac{a}{b}\right)^2 = \frac{a}{b} + 1<math>

and hence

<math>\frac{a}{b} = \varphi.<math>

This definition gives the value of <math>\varphi<math> stated above. Alternatively, some define the number of the golden ratio to be the so-called golden ratio conjugate (also erroneously called the silver ratio or silver mean), <math>\hat\varphi = \frac{1}{\varphi} = \varphi-1<math>. The ratios <math>\varphi:1<math> and <math>1:\hat\varphi<math> are equivalent. Also, some use the symbols <math>\tau<math> or <math>\varphi<math> to designate the number called <math>\hat\varphi<math> here.

The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio". This can be easily visualized using a line that is divided into two segments, as in the diagram.

For those who struggle with algebra but can at least handle the idea of fractions of equal value, the inquiry that leads to the golden number is expressed this way: Is there a number that compares to 1 as 1 compares to the same number minus one? The answer is yes, but there is only one number, and it is the golden number x in this example.

<math>\frac{x}{1} = \frac{1}{x-1},<math>

or, equivalently, the quadratic equation

<math>x^2-x-1=0.\,<math>

This quadratic equation has two roots:

<math>{1+\sqrt{5} \over 2} \approx\ 1.618034,\ \mathrm{and}\ {1-\sqrt{5} \over 2} \approx\ -0.618034. <math>

If a house has a rectangular "golden window" with a length of 1 unit of measurement, then its width is the golden number, about 0.618 of the unit of measurement. If the shorter side of the window is instead determined to have a width of 1 unit of measurement, then its length is 1 plus the golden number, about 1.618 units of measurement.

The golden ratio value <math>\approx\ 1.618034 <math> is the only positive number that is exactly 1 less than its own square.

<math>x^2=x+1\,\!<math>

A startlingly quick proof of irrationality

The relation

<math>\frac{a}{b} = \frac{b}{a-b}<math>

gives a startlingly quick proof that this number is irrational: If a/b is a fraction in lowest terms, then b/(a − b) is in even lower terms — a contradiction.

Properties

φ is an irrational number, and the unique positive real number with

<math>\varphi^{-2} = 2 - \varphi \ <math>

<math>\varphi^{-1} = \varphi - 1 \ <math>

<math>\varphi^0 = 1 <math>

<math>\varphi^1 = \varphi \ <math>

<math>\varphi^2 = \varphi + 1 \ <math>

<math>\varphi^3 = \frac{\varphi + 1}{\varphi - 1}<math>

Alternate forms

The formula <math>\varphi = 1 + 1/\varphi<math> can be expanded recursively to obtain a continued fraction for the golden ratio:

<math>\varphi = [1; 1, 1, 1, ...] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}<math>

and its conjugate:

<math>\hat\varphi = [0; 1, 1, 1, ...] = 0 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}.<math>

Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers.

The equation <math>\varphi^2 = 1 + \varphi<math> likewise produces the continued square root form:

<math>\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}.<math>

Also

<math>\varphi=2\cos 36^\circ=2\cos(\pi/5).\,<math>

which is a consequence of the fact that the length of the diagonal of a regular pentagon is <math>\varphi<math> times the length of its side.

Mathematical uses

"Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."

—Johannes Kepler

The golden rectangle, whose sides a and b stand in the golden ratio, is illustrated below:

Missing image
Golden_rectangle_detailed.png
Image:Golden rectangle detailed.png

If from this rectangle we remove square B with sides of length b, then the remaining rectangle A is again a golden rectangle, since its side ratio is b/(a-b) = a/b = φ. Iterating this construction produces a sequence of progressively smaller golden rectangles; each dividing line marks a point on a logarithmic spiral. The curve of the spiral can be closely approximated by inscribing quarter-circle arcs in each square, while the true logarithmic spiral is expressed θ = (π/2log(φ)) * log r in polar coordinates.

Golden and logarithmic spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a logarithmic spiral. Overlapping portions appear yellow.

Since φ is defined to be the root of a polynomial equation, it is an algebraic number. It can be shown that φ is an irrational number.

The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.

The explicit expression for the Fibonacci sequence involves the golden ratio and its conjugate. Also, the limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio. This means when given a Fibonacci number, multiplying it by φ approximates the next Fibonacci number, and that approximation gets better and better as the Fibonacci numbers get higher. Interestingly enough, if all the approximation errors are added up, they equal φ. Stated mathematically:

<math>\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)| = \varphi.<math>

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

φ = φ,

φ² = φ + 1,

φ³ = 2φ + 1,

φ⁴ = 3φ + 2,

φ⁵ = 5φ + 3,

φ⁶ = 8φ + 5,

...

From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It is also the fundamental unit of the algebraic number field <math>\mathbb{Q}(\sqrt{5})<math> and is a Pisot-Vijayaraghavan number.

The golden ratio has interesting properties when used as the base of a numeral system: see golden mean base.

Aesthetic uses

It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza.

The ancient Greeks already knew the golden ratio from their investigations into geometry, but there is no evidence they thought the number warranted special attention above that for numbers like <math>\pi<math> (Pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. They are, at best, inconclusive. [1] (http://plus.maths.org/issue22/features/golden/) Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. This has encouraged modern artists, architects, and others, during the last 500 years, to incorporate the ratio in their work. The Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio.

It is also claimed that the human body has proportions close to the golden ratio.

In 1509 Luca Pacioli published the Divina Proportione, which explored not only the mathematics of the golden ratio, but also its use in architectural design. This was a major influence on subsequent generations of artists and architects. Leonardo Da Vinci drew the illustrations, leading many to speculate that he himself incorporated the golden ratio into his work, although there is no evidence supporting this.

The Architect Le Corbusier used the golden ratio as the basis of his Modulor system of Architecture.

The ratio is sometimes used in modern man-made constructions, such as stairs and buildings, woodwork, and in paper sizes; however, the series of standard sizes that includes A4 is based on a ratio of <math>\sqrt{2}<math> and not on the golden ratio. The average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [2] (http://arxiv.org/abs/physics/9908036/). Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 percent from the golden ratio.

The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence, making them the closest low integer ratios to the golden ratio. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

The construction of a pentagram is based on the golden ratio. The pentagram can be seen as a geometric shape consisting of 5 straight lines arranged as a star with 5 points. The intersection of the lines naturally divides each length into 3 parts. The smaller part (which forms the pentagon inside the star) is proportional to the longer length (which form the points of the star) by a ratio of 1:1.618... It is thought by some that this fact may be a reason why the ancient philosopher Pythagoras chose the pentagram as the symbol of the secret brotherhood of which he was both leader and founder.

Decimal expansion

Template:OEIS

1.6180339887 4989484820 4586834365 6381177203 0917980576
  2862135448 6227052604 6281890244 9707207204 1893911374
  8475408807 5386891752 1266338622 2353693179 3180060766
  7263544333 8908659593 9582905638 3226613199 2829026788
  0675208766 8925017116 9620703222 1043216269 5486262963
  1361443814 9758701220 3408058879 5445474924 6185695364
  8644492410 4432077134 4947049565 8467885098 7433944221
  2544877066 4780915884 6074998871 2400765217 0575179788
  3416625624 9407589069 7040002812 1042762177 1117778053
  1531714101 1704666599 1466979873 1761356006 7087480710
  1317952368 9427521948 4353056783 0022878569 9782977834
  7845878228 9110976250 0302696156 1700250464 3382437764
  8610283831 2683303724 2926752631 1653392473 1671112115
  8818638513 3162038400 5222165791 2866752946 5490681131
  7159934323 5973494985 0904094762 1322298101 7261070596
  1164562990 9816290555 2085247903 5240602017 2799747175
  3427775927 7862561943 2082750513 1218156285 5122248093
  9471234145 1702237358 0577278616 0086883829 5230459264
  7878017889 9219902707 7690389532 1968198615 1437803149
  9741106926 0886742962 2675756052 3172777520 3536139362
  1076738937 6455606060 5922...

This can also be found fairly easily on your calculator. Simply take the square root of one, then add one to that answer, then take the square root of that result, then add one again, repeat this over and over until the number remains constant. This should give you a pretty good estimate of Phi.

See also

• Golden angle
• Golden function
• Golden rectangle
• Golden section search
• Logarithmic spiral
• Fibonacci number
• Modulor
• Sacred geometry
• The Roses of Heliogabalus

Other meanings

• The Doctrine of the Golden Mean (Template:Zh-cp), the name of a chapter in Li Ji (Template:Zh-cp) is one of the "Four books" of classical Chinese writings.