The Golden Ratio
From Principles of Applied Arts
The golden ratio, also known as the golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina, is an irrational number, approximately 1.618 033 988 749 894 848, that possesses many interesting properties. Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence.
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Definition
Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if
- <math>\frac{a+b}{a} = \frac{a}{b}</math>
where a is the larger part and b is the smaller part. Image:Golden ratio line.png 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, i.e. if
- <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.\qquad\qquad(*)</math>
The Greek letter φ (phi) is conventionally used to denote the size of the larger part when the smaller part is 1, and this number φ is often called "the golden ratio". Thus we have
- <math>\frac{a}{b} = \varphi.</math>
The equation labeled (*) above then becomes
- <math>\varphi^2=\varphi+1\,</math>
or equivalently,
- <math>\varphi^2 - \varphi - 1 \ = \ 0</math>
The solutions of this quadratic equation are
- <math>{1 \pm \sqrt{5} \over 2}.</math>
Since φ is a quantity it must be positive, hence we have
- <math>\varphi = {1 + \sqrt{5} \over 2}\approx\ 1.618 033 988\dots.</math>
History
The golden ratio was first studied by ancient mathematicians because of its frequent appearance in geometry and may have even been understood and used as far back in history as the Egyptians. It is also believed that after tracing the path of Venus in the sky, they found that the ratio of the length of the long arm of the pentagon shape to the length of the shorter arm was 1.618 ... ... More commonly, however, the discovery of the golden ratio is ascribed to the ancient Greeks, and 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 represented by the Greek letter (phi, after Phidias, a sculptor who commonly employed it) or less commonly by τ (tau).
Mathematical uses
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:Image:FakeRealLogSpiral.pngThe limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610 ≈ 1.6180327868852. Alternatingly the approximation to φ is too small and too large, it gets better as the Fibonacci numbers get higher, and:
Furthermore, the successive powers of φ obey the Fibonacci recurrence: φ−2 = − φ + 2, φ−1 = φ − 1, φ0 = 1, φ1 = φ, φ2 = φ + 1, φ3 = 2φ + 1, φ4 = 3φ + 2, φ5 = 5φ + 3, φn = F(n)φ + F(n − 1), ... Because φ is the only positive number that satisfies the identity φn = φn − 1 + φn − 2, any polynomial expression in φ may be decomposed into a linear expression. For example:
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 has been argued this is the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). It is also the fundamental unit of the algebraic number field 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
Image:ParthenonGoldenRatio.png
It has been claimed that the History of Ancient Egypt|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 History of Greece 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. 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. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on 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. It has been suggested for example that Da Vinci's painting of the Mona Lisa employs the Golden Ratio in its geometric equivalents.
The Architect Le Corbusier used the golden ratio as the basis of his Modulor system of Architecture.
Image:Golden-ratio-construction.png Image:250px-Golden section page.png
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 paper size|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. [1]. Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 % from the golden ratio.
The ratios of just intonation 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 shepard tone 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 sixth 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.
Ernő Lendvai (1971) analyses Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.
Image:180px-Divina proportione.png
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 fraternity of which he was both leader and founder.
There is no known general algorithm to arrange a given number of nodes evenly on a sphere (for any of several definitions of "evenly"), but a good approximation can be achieved by dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This approach was used to arrange mirrors on the Starshine 3 satellite.
