Golden ratio
The
golden ratio (
proportio divina or
sectio aurea), also called the
golden mean,
golden section,
golden number or
divine proportion, usually denoted by the
Greek letter Φ (phi), is the
number:
-
Properties
\n\n{| cellpadding="2" cellspacing="0" style="margin:3px; border:3px solid #A0DFFF;width:276px; font-size:95%; font-family:Arial,Helvetica" align="right"\n!bgcolor="#A0DFFF" colspan="3" align="center"|Golden ratio example\n|-\n!bgcolor="#A0DFFF"|
\n|-\n|colspan="3" align="center" bgcolor="#A0DFFF"|The golden ratio can be easily visualized using a line that is divided into two segments a and b. The entire line is to the a segment as a is to the b segment.\n|-\n|}\nΦ is irrational, and the unique positive real number with
-
and the equally interesting properties
-
-
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".
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."
The golden rectangle, whose sides a and b stand in the golden ratio, is illustrated below:
\n |.......... a..........|
+-------------+--------+ -\n | | | .\n | | | .\n | B | A | b\n | | | .\n | | | .\n | | | .\n +-------------+--------+ -
|......b......|..a-b...|\n
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.\nBecause of 1+1/φ = φ, the
continued fraction representation of φ is
-
The number φ turns up frequently in
geometry, in particular in figures involving pentagonal symmetry.\nFor 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.\nAlso, the
limit of ratios of successive terms of the Fibonacci sequence equals the golden ratio. 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.
\n
Aesthetic uses
The ancient Egyptians and
ancient Greeks already knew the number and had identified its nature. They found this mathematical proportion, which they called
The Golden Mean throughout nature, and it impacted their art, architecture,
paideia, and philosophy. The golden mean was found throughout nature, in the structure of nautilus shells, the size of leaves, the branching patterns of trees, and in the human body. The Greeks thought that the golden mean described the dimensions of average, and by inference "ideal", body features such as the face and the torso, and the proportions of arms and legs to the size of the body.
The golden ratio was used as a guide for accurately creating human likenesses in painting and sculpture. Because the ratio was so common in nature, it was considered auspicious and
aesthetically pleasing and was used for other creations, even if not dictated by nature. Many paintings are arranged according to golden ratios; buildings and courtyards were designed with golden rectangles, as were great monuments (e.g., the
Parthenon and the
Great Pyramid at
Giza). The
pentagram so popular among the
Pythagoreans also contains the golden ratio.
The golden mean continues to be used for design. It 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. Recent studies show that the golden ratio continues to play a role in human perception of
beauty, have confirmed its presence in body shapes and faces.
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.
In Nature
\nThe golden ratio turns up in nature as a result of the dynamics of some systems. For instance, in the angular spacing of trees 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
In Computation
It's a good heuristic for any computational problem of the form "we have an arbitrary number of things and we want to put them down without overlapping". Eg: shell sizes in shell sort, inserting items into hash tables.
The golden ratio up to 1024 decimal places
\n 1.6180339887 4989484820 4586834365 6381177203 0917980576\n 2862135448 6227052604 6281890244 9707207204 1893911374\n 8475408807 5386891752 1266338622 2353693179 3180060766\n 7263544333 8908659593 9582905638 3226613199 2829026788\n 0675208766 8925017116 9620703222 1043216269 5486262963\n 1361443814 9758701220 3408058879 5445474924 6185695364\n 8644492410 4432077134 4947049565 8467885098 7433944221\n 2544877066 4780915884 6074998871 2400765217 0575179788\n 3416625624 9407589069 7040002812 1042762177 1117778053\n 1531714101 1704666599 1466979873 1761356006 7087480710\n 1317952368 9427521948 4353056783 0022878569 9782977834\n 7845878228 9110976250 0302696156 1700250464 3382437764\n 8610283831 2683303724 2926752631 1653392473 1671112115\n 8818638513 3162038400 5222165791 2866752946 5490681131\n 7159934323 5973494985 0904094762 1322298101 7261070596\n 1164562990 9816290555 2085247903 5240602017 2799747175\n 3427775927 7862561943 2082750513 1218156285 5122248093\n 9471234145 1702237358 0577278616 0086883829 5230459264\n 7878017889 9219902707 7690389532 1968198615 1437803149\n 9741106926 0886742962 2675756052 3172777520 3536139362\n 1076738937 6455606060 5921...\n
Deriving value from continued fraction
- \n:\n:\n:\n:\n:
Deriving the value from nested radicals
\n:\n:\n:
See also
Other meanings
\nThe Doctrine of the Golden Mean (Zhong1 Yong2, 中庸), the name of a chapter in Li Ji (Li3 ji4, 禮記) is one of the "Four books" of
classical Chinese writings.
External links and references
