 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 3 ] Next Last |
It is usually denoted by the Greek letter φ (phi).
Two quantities are said to be in the golden ratio if "the whole is to the larger as the larger is to the smaller". 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 last quantity is called the golden ratio conjugate (erroneously called the silver ratio or silver mean) and is also represented by .
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
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".
The formula can be expanded recursively to obtain a continued fraction for the golden ratio:
and its conjugate:
The equation 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:
|.......... a..........| +-------------+--------+ - | | | . | | | . | B | A | b | | | . | | | . | | | . +-------------+--------+ - |......b......|..a-b...|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 coordinatesThis article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components)
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 numberAbstract algebra Algebra In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form : a x n + a x n minus;1 + ··· + a x + a 0 where n is a posit. 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 icosahedronAn icosahedron [aiks'hidrn] noun (plural: -drons, -dra [-dr]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangles. Etymology 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedra are located on three orthogonal golden rectangleA golden rectangle is a rectangle with dimensions which are of the Golden Ratio, 1 : φ (i. 6180339887498948. It yields another rectangle with sides of the same proportions when sectioned in a particular manner. That is, sectioned into two shapes: firss.
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 equals the golden ratio. 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.
| Politics | Business | Finance |
 |
 |
 |
|