Golden ratio

 ( Euclidean Plane Geometry ) 

Thus, 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 : $$\backslash varphi\; +\; 1\; =\; \backslash varphi^2$$ which can be rearranged to $$\{\backslash varphi\}^2\; -\; \backslash varphi\; -\; 1\; =\; 0.$$

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.

Ancient Greece mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and . According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreanism. Euclid's Elements () provides several theorem 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 (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the .

Luca Pacioli named his book Divina proportione (1509) 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 ('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 Vitruvius 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 Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) 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". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.

By 1910, inventor Mark Barr began using the Greek alphabet phi () as a symbol for the golden ratio. It has also been represented by tau (), the first letter of the ancient Greek τομή ('cut' or 'section').

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 with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling.

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.

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 Field 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.

$$-\backslash frac\{1\}\{\backslash varphi\}=1-\backslash varphi\; =\; \backslash frac\{1\; -\; \backslash sqrt5\}\{2\}\; =\; -0.618033\backslash dots.$$

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 $$\backslash frac1\backslash varphi\; =\; \backslash varphi\; -\; 1,$$

or its inverse, $$\backslash frac1\{1/\backslash varphi\}\; =\; \backslash frac1\backslash varphi\; +\; 1.$$

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: $$\backslash varphi^n\; =\; \backslash varphi^\{n-1\}\; +\; \backslash varphi^\{n-2\}\; =\; \backslash varphi\; \backslash cdot\; \backslash operatorname\{F\}\_n\; +\; \backslash operatorname\{F\}\_\{n-1\}.$$

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:

= \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5}
= \frac{\varphi^n - (1 - \varphi)^n}{\sqrt5}
= \frac{1}{\sqrt5}\left[\left({ 1+ \sqrt{5} \over 2}\right)^n - \left({ 1- \sqrt{5} \over 2}\right)^n\right],
     
     

= \varphi^n + (- \varphi)^{-n}
= \varphi^n + (1 - \varphi)^n
= \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n .
     
     

  = 2\sqrt{\frac{3-\varphi}{5}}a \approx 1.05146a, \\[5mu]
     

is a Pisot–Vijayaraghavan number.