What is the "Golden Ratio"?

Question

What is the golden ration and how does it relate to beauty.
They say Beauty is in the eye of the beholder, yet the golden ratio claims beauty is a set ratio the nature obeys to.
Such as...
Piramids
Flowers
Characteristics in the human face
And so on

Answer 1

The Golden Ratio is approximately 1 to Phi (approx 1.618) or 0.618 to 1 (mathematically both are the same). Basically it's the ratio you get when you a divide a line AC such that AB/BC = BC/AC.

Are you familiar with the Fibonacci sequence? Each number is the sum of the two preceding numbers:
1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
and so on. So the sequence runs: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc.

If you take any number in the sequence and divide by the preceding number you get a number close to Phi (approx 1.618).
89/55, for example, is 1.6181818181818...

The higher you go up the sequence, the closer you get to an accurate Golden Ratio.

The ratio appears frequently in nature (for example, the ratio of your forearm to your upper arm is about 1 : 1.618), and because of it's aesthetic appeal it has also used frequently artists and architects throughout history. The pyramids and the parthenon use the Golden Ratio.

Answer 2

Piggy backing off of what Ben of Marlow mentioned, that is the ratio of dividing the line such that AB/BC = BC/AC, let us take AB to be one unit and BC to be x. The task is to solve for x; then 1/x is the golden ratio.
By substitution and segment addition, 1/x = x/(1+x). To solve for x, cross multiply then put all terms on one side of an equation, forming a quadratic:

x^2 - x - 1 = 0

The solution to this (by applying the quadratic formula) is

x = (1 + sqrt(5))/2

(we can rule out the negative solution since we're dealing with a line segment).

The number above has been regarded as a magical number. For instance, the golden ratio appears in a pentagram (if you look at how each long line segment is divided), and the pentagram has been thought to possess magical properties.

Answer 3

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes . The designations "phi" (for ) and "Phi" (for the golden ratio conjugate ) are sometimes also used (Knott).

The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6). Similarly, the alternate notation is an abbreviation of the Greek tome, meaning "to cut."

In the Season 1 episode "Sabotage" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the golden ratio is found in the pyramids of Giza and the Parthenon at Athens. Similar, the character Robert Langdon in the novel The Da Vinci Code makes similar such statements (Brown 2003, pp. 93-95). However, claims of the significance of the golden ratio appearing prominently in art, architecture, sculpture, anatomy, etc., tend to be greatly exaggerated.

has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers.


Given a rectangle having sides in the ratio , is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle, and successive points dividing a golden rectangle into squares lie on a logarithmic spiral. This figure is known as a whirling square.

The legs of a golden triangle (an isosceles triangle with a vertex angle of ) are in a golden ratio to its base and, in fact, this was the method used by Pythagoras to construct . The ratio of the circumradius to the length of the side of a decagon is also ,

(1)

Bisecting a (schematic) Gaullist cross also gives a golden ratio (Gardner 1961, p. 102).


Euclid ca. 300 BC defined the "extreme and mean ratios" on a line segment as the lengths such that

(2)

(Livio 2002, pp. 3-4). Plugging in,

(3)

and clearing denominators gives

(4)

(Incidentally, this means that is a algebraic number of degree 2.) So, using the quadratic equation and taking the positive sign (since the figure is defined so that ),

(5)
(6)

(Sloane's A001622). In an apparent blatant misunderstanding of the difference between an exact quantity and an approximation, the character Robert Langdon in the novel The Da Vinci Code incorrectly defines the golden ratio as exactly 1.618 (Brown 2003, pp. 93-95).

Exact trigonometric formulas for include

(7)
(8)
(9)

The golden ratio is given by the infinite series

(10)

(B. Roselle). Another fascinating connection with the Fibonacci numbers is given by the infinite series

(11)

A representation in terms of a nested radical is

(12)

(Livio 2002, p. 83).

is the "most" irrational number because it has a continued fraction representation

(13)

(Sloane's A000012; Williams 1979, p. 52; Steinhaus 1999, p. 45; Livio 2002, p. 84). This means that the convergents are given by the quadratic recurrence equation

(14)

with , which has solution

(15)

where is the th Fibonacci number. This gives the first few convergents as 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, ... (Sloane's A000045 and A000045), which are good to 0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, ... (Sloane's A114540) decimal digits, respectively.

As a result,

(16)

as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62; Livio 2002, p. 101).

The golden ratio also satisfies the recurrence relation

(17)

Taking gives the special case

(18)

Treating (◇) as a linear recurrence equation

(19)

in , setting and , and solving gives

(20)

as expected. The powers of the golden ratio also satisfy

(21)

where is a Fibonacci number (Wells 1986, p. 39).

The sine of certain complex numbers involving gives particularly simple answers, for example

(22)
(23)

(D. Hoey, pers. comm.). A curious (although not particularly useful) approximation due to D. Barron is given by

(24)

where is Catalan's constant and is the Euler-Mascheroni constant, which is good to two digits.


In the figure above, three triangles can be inscribed in the rectangle of arbitrary aspect ratio such that the three right triangles have equal areas by dividing and in the golden ratio. Then

(25)
(26)
(27)

which are all equal.

The substitution map

(28)
(29)

gives

(30)

giving rise to the sequence

(31)

(Sloane's A003849). Here, the zeros occur at positions 1, 3, 4, 6, 8, 9, 11, 12, ... (Sloane's A000201), and the ones occur at positions 2, 5, 7, 10, 13, 15, 18, ... (Sloane's A001950). These are complementary Beatty sequences generated by and . This sequence also has many connections with the Fibonacci numbers. It is plotted above (mod 2) as a recurrence plot.


Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . As can be seen from the plots above, the regularity in the continued fraction of means that is one of a set of numbers of measure 0 whose continued fraction sequences do not converge to the Khinchin constant or the Khinchin-Lévy constant.

The golden ratio has Engel expansion 1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, ... (Sloane's A028259).


Steinhaus (1983, pp. 48-49) considers the distribution of the fractional parts of in the intervals bounded by 0, , , ..., , 1, and notes that they are much more uniformly distributed than would be expected due to chance (i.e., is close to an equidistributed sequence). In particular, the number of empty intervals for , 2, ..., are a mere 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, ... (Sloane's A036414). The values of for which no bins are left blank are then given by 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, ... (Sloane's A036415). Steinhaus (1983) remarks that the highly uniform distribution has its roots in the continued fraction for .

The sequence , of power fractional parts, where is the fractional part, is equidistributed for almost all real numbers , with the golden ratio being one exception.

Salem showed that the set of Pisot numbers is closed, with the smallest accumulation point of the set (Le Lionnais 1983).

Answer 4

1.618- the ratio rife in nature. Bees have a natural equilibrium with a ratio of males to females in the hive 1:1.618. It is also used in architecture to make building aesthetically pleasing. If you measure the length of your finger bottom to tip, then knuckle to tip, they are in the ratio 1;1.618

Answer 5

It is called golden ratio, not ration.
Ration is what you allow some one to eat. Ratio is what you get when you divide one number by another. What are they teaching in schools these days?

Answer 6

This features heavily in the book The Da Vinci Code, or if you aren't quite up to understanding the plot, you can see the film.

Anyway, the book gives quite alot of details, as well as examples in nature.

Answer 7

the golden ratio = 1.61803399
This is mathematical expressionism a la Fabonacci.
It describes the way the natural world is wraped,constructed and naturally fabricated.Amazing stuff!!!!!

Answer 8

In printing and typography it is 3/5; that is place the center of your composition 3/5 up the page so it will appear visually centered. Can't believe this will help but you asked.

Answer 9

They call it the 'golden ratio' because our body are assembled in accordance to it. If you take the length of you leg and divide it by the length of from your toes to your knee, you get 1.618. Your arm to the length to your elbow- 1.618. Your whole body divided by the length to your waist- 1.618. Crazy stuff i know.

Answer 10

its 1 : 1.618