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The golden ratio is a number, approximately 1.618, that possesses many interesting properties. It was studied by ancient mathematicians due to its frequent appearance in geometry. Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.
It is usually denoted by the Greek letter Φ (phi).
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The name "golden ratio" first seemed to have been used in the form sectio aurea, "golden section", by Leonardo da Vinci. The use of the symbol Φ to represent the golden ratio was invented by the American mathematician Mark Barr and taken from the first Greek letter in the name of the Greek sculptor Phidias, who was long believed to have used the golden ratio in his designs.
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
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:
After simple algebraic manipulations (multiply the first equation with a/b or the second equation with (a − b)/b), both of these equations are seen to be equivalent to
and hence
This definition gives the value of φ stated above. Alteratively, 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), . The ratios φ:1 and are equivalent. Also, some use the symbols τ or φ to designate the number called 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.
Φ is an irrational number, and the unique positive real number with
The formula φ = 1 + 1 / φ can be expanded recursively to obtain a continued fraction for the golden ratio:
and its conjugate:
The equation φ2 = 1 + φ likewise produces the continued square root form:
"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."
The golden rectangle, whose sides a and b stand in the golden ratio, is illustrated below:
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. By iterating this construction, one can produce a sequence of progressively smaller golden rectangles; by drawing a quarter circle into each of the discarded squares, one obtains a figure which closely resembles the logarithmic spiral θ = (π/2log(φ)) * log r. (see polar coordinates)
The green spiral is made from quarter circle pieces as described above, the red spiral is a real logarithmic spiral. The similarity between the spirals should be noticeable. (If you
instead only see a yellow spiral, look very carefully, there are actually two different spirals in
the image.)
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 approximations are added up, they equal Φ. Stated mathematically:
Furthermore, the successive powers of Φ obey the Fibonacci recurrence. 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 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.
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 π (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.
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 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 and not on the golden ratio. It's also interesting to note that the average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [2] (http://arxiv.org/abs/physics/9908036/)
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 golden ratio turns up in nature as a result of the dynamics of some systems - for instance, in the angular spacing of tree limbs around a trunk, or sunflower seeds. In both cases, the problem is "wedge this next one into the biggest available space".
You can draw a nice sunflower by plotting the points
The shape of the shell of the chambered nautilus (Nautilus pompilius) is often claimed to be related to the golden ratio. (Please see discussion about the "Golden ratio" page!)
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