In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two quantities are in the golden ratio if their
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
is the same as the ratio of their
sum to the larger of the two quantities. Expressed algebraically, for quantities
and
with
,
where the Greek letter
phi
Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
( or
) denotes the golden ratio. The constant
satisfies the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
and is an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
with a value of
The golden ratio was called the extreme and mean ratio by
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
,
and the divine proportion by
Luca Pacioli
Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
,
and also goes by several other names.
Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
's diagonal to its side and thus appears in the
construction
Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and com ...
of the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
and
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
. A
golden rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity ...
—that is, a rectangle with an aspect ratio of
—may be cut into a square and a smaller rectangle with the same
aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as
financial market
A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets ...
s, in some cases based on dubious fits to data.
The golden ratio appears in some
patterns in nature
Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foa ...
, including the
spiral arrangement of leaves and other parts of vegetation.
Some 20th-century
artist
An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating an art. The common usage in both everyday speech and academic discourse refers to a practitioner in the visual arts only. However, th ...
s and
architect
An architect is a person who plans, designs and oversees the construction of buildings. To practice architecture means to provide services in connection with the design of buildings and the space within the site surrounding the buildings that h ...
s, including
Le Corbusier
Charles-Édouard Jeanneret (6 October 188727 August 1965), known as Le Corbusier ( , , ), was a Swiss-French architect, designer, painter, urban planner, writer, and one of the pioneers of what is now regarded as modern architecture. He was ...
and
Salvador Dalí
Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (; ; ; 11 May 190423 January 1989) was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, and the striking and bizarr ...
, have proportioned their works to approximate the golden ratio, believing it to be
aesthetically
Aesthetics, or esthetics, is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines aesthetic values, often expressed th ...
pleasing. These uses often appear in the form of a golden rectangle.
Calculation
Two quantities
and
are in the ''golden ratio''
if
One method for finding
's closed form starts with the left fraction. Simplifying the fraction and substituting the reciprocal
,
Therefore,
Multiplying by
gives
which can be rearranged to
The
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
yields two solutions:
Because
is a ratio between positive quantities,
is necessarily the positive root.
The negative root is in fact the negative inverse
, which shares many properties with the golden ratio.
History
According to
Mario Livio
Mario Livio (born June 19, 1945) is an Israeli-American astrophysicist and an author of works that popularize science and mathematics. For 24 years (1991-2015) he was an astrophysicist at the Space Telescope Science Institute, which operates th ...
,
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
mathematicians first studied the golden ratio because of its frequent appearance in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
s and
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
s. According to one story, 5th-century BC mathematician
Hippasus
Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
discovered that the golden ratio was neither a whole number nor a fraction (an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
), surprising
Pythagoreans
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
.
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''
Elements'' () provides several
propositions
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the n ...
and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:
The golden ratio was studied peripherally over the next millennium.
Abu Kamil
Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ ( Latinized as Auoquamel, ar, أبو كامل شجاع بن أسلم بن محمد بن شجاع, also known as ''Al-ḥāsib al-miṣrī''—lit. "the Egyptian reckoner") (c. 850 – ...
(c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
(Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s.
Luca Pacioli
Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
named his book ''
Divina proportione
''Divina proportione'' (15th century Italian for ''Divine proportion''), later also called ''De divina proportione'' (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da V ...
'' (
1509) after the ratio; the book, largely plagiarized from
Piero della Francesca
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...
, explored its properties including its appearance in some of the
Platonic solids
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
.
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
, who illustrated Pacioli's book, called the ratio the ''sectio aurea'' ('golden section').
Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the
Vitruvian
The ''Vitruvian Man'' ( it, L'uomo vitruviano; ) is a drawing by the Italian Renaissance artist and scientist Leonardo da Vinci, dated to . Inspired by the writings by the ancient Roman architect Vitruvius, the drawing depicts a nude man in two s ...
system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as
Rafael Bombelli
Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.
He was the one who finally manag ...
solved geometric problems using the ratio.
German mathematician Simon Jacob (d. 1564) noted that
consecutive Fibonacci numbers converge to the golden ratio;
this was rediscovered by
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
in 1608. The first known
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
approximation of the (inverse) golden ratio was stated as "about
" in 1597 by
Michael Maestlin
Michael Maestlin (also Mästlin, Möstlin, or Moestlin) (30 September 1550 – 26 October 1631) was a German astronomer and mathematician, known for being the mentor of Johannes Kepler. He was a student of Philipp Apian and was known as the tea ...
of the
University of Tübingen
The University of Tübingen, officially the Eberhard Karl University of Tübingen (german: Eberhard Karls Universität Tübingen; la, Universitas Eberhardina Carolina), is a public research university located in the city of Tübingen, Baden-Wü ...
in a letter to Kepler, his former student.
The same year, Kepler wrote to Maestlin of the
Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...
, which combines the golden ratio with the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
. Kepler said of these:
18th-century mathematicians
Abraham de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He moved ...
,
Nicolaus I Bernoulli
Nicolaus Bernoulli (also spelled Nicolas or Nikolas; 21 October 1687, Basel – 29 November 1759, Basel) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.
Biography
He was the son of Nicolaus Bern ...
, and
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by
Jacques Philippe Marie Binet
Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical founda ...
, for whom it was named "Binet's formula".
Martin Ohm
Martin Ohm (May 6, 1792 in Erlangen – April 1, 1872 in Berlin) was a German mathematician and a younger brother of physicist Georg Ohm.
Biography
He earned his doctorate in 1811 at Friedrich-Alexander-University, Erlangen-Nuremberg where his ...
first used the German term ''goldener Schnitt'' ('golden section') to describe the ratio in 1835.
James Sully
James Sully (3 March 1842 – 1 November 1923) was an English psychologist.
Biography
James Sully was born at Bridgwater, Somerset, the son of J. W. Sully, a liberal Baptist merchant and ship-owner. He was educated at the Independent Colle ...
used the equivalent English term in 1875.
By 1910, inventor
Mark Barr
James Mark McGinnis BarrFull name as listed in (May 18, 1871December 15, 1950) was an electrical engineer, physicist, inventor, and polymath known for proposing the standard notation for the golden ratio. Born in America, but with English citi ...
began using the
Greek letter
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
Phi
Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
as a
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
for the golden ratio. It has also been represented by
tau
Tau (uppercase Τ, lowercase τ, or \boldsymbol\tau; el, ταυ ) is the 19th letter of the Greek alphabet, representing the voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300.
The name in English ...
the first letter of the
ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
τομή ('cut' or 'section').
The
zome construction system, developed by
Steve Baer
Steve Baer (born 1938) is an American inventor and pioneer of passive solar technology. Baer helped popularize the use of zomes. He took a number of solar power patents, wrote a number of books and publicized his work. Baer served on the board ...
in the late 1960s, is based on the
symmetry system of the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
/
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, and uses the golden ratio ubiquitously. Between 1973 and 1974,
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...
developed
Penrose tiling
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without r ...
, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.
This gained in interest after
Dan Shechtman
Dan Shechtman ( he, דן שכטמן; born January 24, 1941)[Dan Shechtman](_blank)
. (PDF). Retri ...
's Nobel-winning 1982 discovery of
quasicrystal
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical cr ...
s with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.
Mathematics
Irrationality
The golden ratio is an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms
Recall that:
If we call the whole
and the longer part
then the second statement above becomes
To say that the golden ratio
is rational means that
is a fraction
where
and
are integers. We may take
to be in
lowest terms
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). I ...
and
and
to be positive. But if
is in lowest terms, then the equally valued
is in still lower terms. That is a contradiction that follows from the assumption that
is rational.
By irrationality of
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the
closure of rational numbers under addition and multiplication. If
is rational, then
is also rational, which is a contradiction if it is already known that the square root of all non-
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s are irrational.
Minimal polynomial
The golden ratio is also an
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
and even an
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
. It has
minimal polynomial
This
quadratic polynomial
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...
has two
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
,
and
The golden ratio is also closely related to the polynomial
which has roots
and
As the root of a quadratic polynomial, the golden ratio is a
constructible number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is con ...
.
Golden ratio conjugate and powers
The
conjugate root to the minimal polynomial
is
The absolute value of this quantity corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length,
).
This illustrates the unique property of the golden ratio among positive numbers, that
or its inverse:
The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with
:
The sequence of powers of
contains these values
more generally,
any power of
is equal to the sum of the two immediately preceding powers:
As a result, one can easily decompose any power of
into a multiple of
and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of
:
If
then:
Continued fraction and square root
The formula
can be expanded recursively to obtain a
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
for the golden ratio:
It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
The
convergents of these continued fractions
... or
are ratios of successive
Fibonacci numbers
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the
Hurwitz inequality for
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
s, which states that for every irrational
, there are infinitely many distinct fractions
such that,
This means that the constant
cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such
Lagrange numbers.
A
continued square root form for
can be obtained from
, yielding:
Relationship to Fibonacci and Lucas numbers
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s and
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci nu ...
s have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence
:
The sequence of Lucas numbers (not to be confused with the generalized
Lucas sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation
: x_n = P \cdot x_ - Q \cdot x_
where P and Q are fixed integers. Any sequence satisfying this recu ...
s, of which this is part) is like the Fibonacci sequence, in-which each term is the sum of the previous two, however instead starts with
:
Exceptionally, the golden ratio is equal to the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:
In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates
.
For example,
and
These approximations are alternately lower and higher than
and converge to
as the Fibonacci and Lucas numbers increase.
Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
s for the Fibonacci and Lucas sequences that involve the golden ratio are:
Combining both formulas above, one obtains a formula for
that involves both Fibonacci and Lucas numbers:
Between Fibonacci and Lucas numbers one can deduce
which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the
square root of five:
Indeed, much stronger statements are true:
:
,
:
.
These values describe
as a
fundamental unit of the
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
.
Successive powers of the golden ratio obey the Fibonacci
recurrence
Recurrence and recurrent may refer to:
*''Disease recurrence'', also called relapse
*''Eternal recurrence'', or eternal return, the concept that the universe has been recurring, and will continue to recur, in a self-similar form an infinite number ...
, i.e.
The reduction to a linear expression can be accomplished in one step by using:
This identity allows any polynomial in
to be reduced to a linear expression, as in:
Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by
infinite summation:
In particular, the powers of
themselves round to Lucas numbers (in order, except for the first two powers,
and
, are in reverse order):
and so forth.
The Lucas numbers also directly generate powers of the golden ratio; for
:
Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of ''third'' consecutive Fibonacci numbers equals a Lucas number, that is
; and, importantly, that
.
Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the
golden spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter turn it makes.
Approximations of the golden spira ...
(which is a special form of a
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
) using quarter-circles with radii from these sequences, differing only slightly from the ''true'' golden logarithmic spiral. ''Fibonacci spiral'' is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.
Geometry
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
, and extends to form part of the coordinates of the vertices of a
regular dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
, as well as those of a
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It i ...
. It features in the
Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...
and
Penrose tilings too, as well as in various other
polytopes
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...
.
Construction
Dividing by interior division
# Having a line segment
construct a perpendicular
at point
with
half the length of
Draw the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
# Draw an arc with center
and radius
This arc intersects the hypotenuse
at point
# Draw an arc with center
and radius
This arc intersects the original line segment
at point
Point
divides the original line segment
into line segments
and
with lengths in the golden ratio.
Dividing by exterior division
# Draw a line segment
and construct off the point
a segment
perpendicular to
and with the same length as
# Do bisect the line segment
with
# A circular arc around
with radius
intersects in point
the straight line through points
and
(also known as the extension of
). The ratio of
to the constructed segment
is the golden ratio.
Application examples you can see in the articles
Pentagon with a given side length,
Decagon with given circumcircle and Decagon with a given side length.
Both of the above displayed different
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s produce
geometric construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
s that determine two aligned
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s where the ratio of the longer one to the shorter one is the golden ratio.
Golden angle
When two angles that make a full circle have measures in the golden ratio, the smaller is called the ''golden angle'', with measure
This angle occurs in
patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.
Golden spiral
Logarithmic spirals are
self-similar
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
spirals where distances covered per turn are in
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
. Importantly, isosceles
golden triangles can be encased by a golden logarithmic spiral, such that successive turns of a spiral generate new golden triangles inside. This special case of logarithmic spirals is called the ''golden spiral'', and it exhibits continuous growth in golden ratio. That is, for every
turn, there is a growth factor of
. As mentioned above, these ''golden spirals'' can be approximated by quarter-circles generated from Fibonacci and Lucas number-sized squares that are tiled together. In their exact form, they can be described by the
polar equation
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
with
:
As with any logarithmic spiral, for
with
at
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s:
Its
polar slope can be calculated using
alongside
from above,
It has a complementary angle,
:
Golden spirals can be symmetrically placed inside pentagons and pentagrams as well, such that fractal copies of the underlying geometry are reproduced at all scales.
In triangles, quadrilaterals, and pentagons
=Odom's construction
=
George Odom found a construction for
involving an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
: if an equilateral triangle is
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion.
=Kepler triangle
=
The ''Kepler triangle'', named after
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
, is the unique
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
with sides in
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
:
The Kepler triangle can also be understood as the right triangle formed by three squares whose areas are also in golden geometric progression
.
Fittingly, the
Pythagorean means for
are precisely
,
, and
. It is from these ratios that we are able to geometrically express the fundamental defining quadratic polynomial for
with the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
; that is,
.
The
inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
of an isosceles triangle is greatest when the triangle is composed of two
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
Kepler triangles, such that their bases lie on the same
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
Also, the isosceles triangle of given
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
with the largest possible
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line of ...
is one from two mirrored Kepler triangles.
For a Kepler triangle with smallest side length
, the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
and
acute
Acute may refer to:
Science and technology
* Acute angle
** Acute triangle
** Acute, a leaf shape in the glossary of leaf morphology
* Acute (medicine), a disease that it is of short duration and of recent onset.
** Acute toxicity, the adverse eff ...
internal angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s are:
=Golden triangle
=
A ''golden triangle'' is characterized as an
isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
with the property that
bisecting the angle
produces new acute and obtuse isosceles triangles
and
that are
similar to the original, as well as in
leg
A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element ca ...
to
base length ratios of
and
, respectively.
The acute isosceles triangle is sometimes called a ''sublime triangle'', and the ratio of its base to its equal-length sides is
.
Its apex angle
is equal to:
Both base angles of the isosceles golden triangle equal
degrees each, since the sum of the angles of a triangle must equal
degrees. It is the only triangle to have its three angles in
ratio.
A
regular pentagram contains five acute sublime triangles, and a
regular decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°.
A self-intersecting ''regular decagon'' i ...
contains ten, as each two vertices connected to the center form acute golden triangles.
The obtuse isosceles triangle is sometimes called a ''golden gnomon'', and the ratio of its base to its other sides is the reciprocal of the golden ratio,
.
The measure of its apex angle
is:
Its two base angles equal
each. It is the only triangle whose internal angles are in
ratio. Its base angles, being equal to
, are the same measure as that of the acute golden triangle's apex angle. Five golden gnomons can be created from adjacent sides of a pentagon whose non-coincident vertices are joined by a diagonal of the pentagon.
Appropriately, the ratio of the area of the obtuse golden gnomon to that of the acute sublime triangle is in
golden ratio. Bisecting a base angle inside a sublime triangle produces a golden gnomon, and another a sublime triangle. Bisecting the apex angle of a golden gnomon in
ratio produces two new golden triangles, too. Golden triangles that are decomposed further like this into pairs of isosceles and obtuse golden triangles are known as ''Robinson triangles.''
=Golden rectangle
=
The golden ratio proportions the adjacent side lengths of a ''golden rectangle'' in
ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in
ratio. They can be generated by ''golden spirals'', through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
as well as in the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
(see section below for more detail).
=Golden rhombus
=
A ''golden rhombus'' is a
rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
whose diagonals are in proportion to the golden ratio, most commonly
.
For a rhombus of such proportions, its acute angle and obtuse angles are:
The lengths of its short and long diagonals
and
, in terms of side length
are:
Its area, in terms of
,and
:
Its
inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
, in terms of side
:
Golden rhombi feature in the
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
(see section below). They also are found in the
golden rhombohedron, the
Bilinski dodecahedron
In geometry, the Bilinski dodecahedron is a Convex set, convex polyhedron with twelve Congruence (geometry), congruent golden rhombus faces. It has the same topology but a different geometry than the face-transitive rhombic dodecahedron. It is a ...
,
and the
rhombic hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach.
It is topologically ident ...
.
=Pentagon and pentagram
=
In a
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying
Ptolemy's theorem
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician ...
to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are
and short edges are
then Ptolemy's theorem gives
which yields,
The diagonal segments of a pentagon form a
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
, or five-pointed
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
, whose geometry is quintessentially described by
. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is
as the four-color illustration shows.
A pentagram has ten
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s: five are
acute
Acute may refer to:
Science and technology
* Acute angle
** Acute triangle
** Acute, a leaf shape in the glossary of leaf morphology
* Acute (medicine), a disease that it is of short duration and of recent onset.
** Acute toxicity, the adverse eff ...
''sublime triangles'', and five are
obtuse ''golden gnomons.'' In all of them, the ratio of the longer side to the shorter side is
These can be decomposed further into pairs of golden Robinson triangles, which become relevant in
Penrose tilings.
Otherwise, pentagonal and pentagrammic geometry permits us to calculate the following values for
:
=Penrose tilings
=
The golden ratio appears prominently in the ''Penrose tiling'', a family of
aperiodic tiling
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- period ...
s of the plane developed by
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...
, inspired by
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.
Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
*Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.
*The kite and dart Penrose tiling uses
kites
A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...
with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.
*The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. Again, these rhombi can be decomposed into golden Robinson triangles. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals
, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of these two tiles are in the golden ratio to each other.
In the dodecahedron and icosahedron
The
regular dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges ...
and its
dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
are
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s whose dimensions are related to the golden ratio. An icosahedron is made of
regular pentagonal faces, whereas the icosahedron is made of
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
s; both with
edges.
For a dodecahedron of side
, the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of a circumscribed and inscribed sphere, and
midradius are (
,
and
, respectively):
While for an icosahedron of side
, the radius of a circumscribed and inscribed sphere, and
midradius are:
The volume and surface area of the dodecahedron can be expressed in terms of
:
As well as for the icosahedron:
These geometric values can be calculated from their
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, which also can be given using formulas involving
. The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the
cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
s of:
Sets of three golden rectangles intersect
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
ly inside dodecahedra and icosahedra, forming
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
.
In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain
vertices of the icosahedron, or equivalently, intersect the centers of
of the dodecahedron's faces.
A
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
can be
inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...
in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is
times that of the dodecahedron's.
In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in
ratio. On the other hand, the
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's
vertices touch the
edges of an octahedron at points that divide its edges in golden ratio.
Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the
compound of five cubes
The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.
It is one of five regular compounds, and dual to the compound of five octahedra. It can be seen as a faceting of a regula ...
,
compound of five octahedra
The compound of five octahedra is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876. It is unique among the regula ...
,
compound of five tetrahedra
The polyhedral compound, compound of five tetrahedron, tetrahedra is one of the five regular polyhedral compounds. This Polyhedral compound, compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hes ...
, the
compound of ten tetrahedra
The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.
It can be seen as a f ...
,
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
,
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...
,
truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. ...
,
truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
Geometric relations
This polyhedron can be formed from a regular dodecahedron by truncat ...
, and
rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
It has 20 regular triangular faces, 30 square (geometry), square face ...
,
rhombic enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 Rhombus, rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedr ...
, and
Kepler-Poinsot polyhedra, and
rhombic hexecontahedron
In geometry, a rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry. It was described mathematically in 1940 by Helmut Unkelbach.
It is topologically ident ...
. In four dimensions, the dodecahedron and icosahedron appear as faces of the
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
and
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from " ...
, which again have dimensions related to the golden ratio.
Other properties
The golden ratio's ''decimal expansion'' can be calculated via root-finding methods, such as
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
or
Halley's method
In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley.
The algorithm is second in the class of Householder's m ...
, on the equation
or on
(to compute
first). The time needed to compute
digits of the golden ratio using Newton's method is essentially
, where
is
the time complexity of multiplying two
-digit numbers.
This is considerably faster than known algorithms for
and
. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers
and
each over
digits, yields over
significant digits of the golden ratio. The decimal expansion of the golden ratio
has been calculated to an accuracy of ten trillion digits.
The golden ratio and inverse golden ratio
have a set of symmetries that preserve and interrelate them. They are both preserved by the
fractional linear transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfor ...
s
– this fact corresponds to the identity and the definition quadratic equation.
Further, they are interchanged by the three maps
– they are reciprocals, symmetric about
and (projectively) symmetric about
More deeply, these maps form a subgroup of the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
isomorphic to the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
on
letters,
corresponding to the
stabilizer of the set
of
standard points on the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
, and the symmetries correspond to the quotient map
– the subgroup
consisting of the identity and the
-cycles, in
cycle notation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
fixes the two numbers, while the
-cycles
interchange these, thus realizing the map.
In the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, the fifth
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
(for an integer
) satisfying
are the vertices of a pentagon. They do not form a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
:
with and (usual) integers. When algebrai ...
s, however the sum of any fifth root of unity and its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
,
''is'' a quadratic integer, an element of
Specifically,
This also holds for the remaining tenth roots of unity satisfying
For the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, the only solutions to the equation
are
and
.
When the golden ratio is used as the base of a
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
(see
golden ratio base
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, ...
, sometimes dubbed ''phinary'' or
''-nary''),
quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
:
with and (usual) integers. When algebrai ...
s in the ring