Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with, is in a golden ratio to if where the Greek letter phi denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of
The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names.
Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—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 markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.
Calculation
Two quantities and are in the golden ratio ifThus, if we want to find, we may use that the definition above holds for arbitrary ; thus, we just set, in which case and we get the equation
which becomes a quadratic equation after multiplying by :
which can be rearranged to
The quadratic formula 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,Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction, surprising Pythagoreans. Euclid's Elements provides several propositions 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 employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci , who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.
Luca Pacioli named his book Divina proportione after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea. 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 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 solved geometric problems using the ratio.
German mathematician noted that [|ratios of consecutive Fibonacci numbers converge to the golden ratio]; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:
Eighteenth-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler 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, for whom it was named "Binet's formula". In 1789, Johann Samuel Traugott Gehler made the earliest known use of the term 'golden section’ in his popular dictionary of physical sciences, Physikalisches Wörterbuch, referring to it as ‘güldnen Schnitt ’. James Sully used the equivalent English term in 1875.
By 1910, inventor Mark Barr began using the Greek letter phi as a symbol for the golden ratio. It has also been represented by tau, the first letter of the ancient Greek τομή.
File:Dan Shechtman in 1985.jpg|thumb|right|Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.
The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, 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's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling.
Mathematics
Irrationality
The golden ratio is an irrational number. Below are two short proofs of irrationality:Contradiction from an expression in lowest terms
This is a proof by infinite descent. 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 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 the square root of 5
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 assumed to be rational, then, the square root of, must also be rational. This is a contradiction, as the square roots of all non-square natural numbers are irrational.Minimal polynomial
Since the golden ratio is a root of a polynomial with rational coefficients, it is an algebraic number. Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is This quadratic polynomial has two roots, and. Because the leading coefficient of this polynomial is 1, both roots are algebraic integers. The golden ratio is also closely related to the polynomial, which has roots and.The golden ratio is a fundamental unit of the quadratic field, sometimes called the golden field. In this field, any element can be written in the form, with rational coefficients and ; such a number has norm. Other units, with norm, are the positive and negative powers of. The quadratic integers in this field, which form a ring, are all numbers of the form where and are integers.
As the root of a quadratic polynomial, the golden ratio is a constructible number.
Golden ratio conjugate and powers
The conjugate root to the minimal polynomial isThe absolute value of this quantity corresponds to the length ratio taken in reverse order.
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 simple continued fraction 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. 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 approximations, 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
s and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term is equal to the sum of the preceding two terms and, starting with the base sequence as the 0th and 1st terms and :The sequence of Lucas numbers is like the Fibonacci sequence, in that each term is the sum of the previous two terms and, however instead starts with as the 0th and 1st terms and :
Exceptionally, the golden ratio is equal to the limit 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,
These approximations are alternately lower and higher than, and converge to as the Fibonacci and Lucas numbers increase.
Closed-form expressions 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:
Successive powers of the golden ratio obey the Fibonacci recurrence,.
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 :
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 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.