What is the golden Ratio...?

Simple answer: Any non-square rectangle can have a line drawn within it parallel to the short side that partitions it into a square and a smaller rectangle. If the long side of the rectangle is a and the short side is b, the new rectangle will have dimensions of b x (a-b). If this is not obvious to you, remember that one side of the rectangle is the side of the square (which has dimensions b x b), and the other is the part of the long side which is not occupied by the square (thus having a length equal to the length of the long side minus the length of the square).

Now, some rectangles, when so partitioned, have the property that the new rectangle formed by the partition will have the same ratio between the lengths of its sides as the original rectangle. That is, a/b = b/(a-b). Such rectangles are called golden rectangles, and the ratio of the sides of a golden rectangle is called the golden ratio.

The golden ratio is easily shown to be both irrational and algebraic. Irrationality is proven by contradiction: note that all rational numbers can be represented in lowest terms - that is, as the ratio of two integers that have no common factors other than 1. However, the golden ratio cannot be represented in lowest terms, since if some ratio a/b were a representation in lowest terms, b/(a-b) would be a valid representation in even lower terms, which contradicts our statement that a/b is in lowest terms. Q.E.D.

Computing the golden ratio is done by solving the equation we provided: a/b = b/(a-b). We arbitrarily choose b to be 1 - we could let b be anything, but letting b be 1 makes the ratio a/b much easier to compute, since it will simply be equal to a. Thus we have a=1/(a-1). Multiplying, we get a(a-1)=1 and from that a² - a = 1 Putting this in standard form, we have a² - a - 1 = 0, which shows that the golden ratio is algebraic, since our formula for computing it IS a polynomial - this stands in contrast to many other fundamental constants like pi and e, which cannot be expressed as a root of any polynomial of any degree (that is, they are non-algebraic).

Completing the square then yeilds:

a² - a - 1 = 0

a² - a + 1/4 = 5/4

(a - 1/2)² = 5/4

a - 1/2 = ±√(5)/2

a = (1 ± √5)/2

However, since the golden ratio is a ratio of lengths, and all lengths are positive, we know that it cannot be (1 - √5)/2, since that is negative. Thus the golden ratio is (1 + √5)/2 exactly.

You 've picked a great subject and with some work you will probably create a presentation which will amaze your class and yourself. I intend to help you doing so by providing the following remarks and WEB LINKS:

The golden ratio is equal the ratio between the number phi= (1+sqrt5)/2 and the number 1 (approximately 1.61803). Thus it can be expressed by the number phi. I referred to it as a ratio of two numbers on purpose, because in the cases where it is important it occures as a ratio.

It has been observed that objects characterized by this ratio appear to he human eye as extremely well proportioned. That is why ancient Greek sculptors and architects made use of it quiet frequently.

The golden ratio can be observed in nature in the following circumstances (and many, many more, CHECK THE LINKS ON THE WEBSITES BELOW, YOU WILL FIND MATERIAL FOR MORE HAN 10 PRESENTATIONS):

----The females of ANY honeybee comunity are phi times more than the males (in very good approximation. It can't be exact because phi is irrational. Instead their numbers are adjascend members of the Fibonacci series, which we will discuss later).

----Phi appears to be the ratio of the diameters of spirals of ALL nautiluses. Same holds for certain spirals created by ANY sunflower's seeds.

As a part of your presentation (before explaining it) you can ask your class to measure the distances from fingertip to shoulder and from fingertip to elbow. By dividing you will obtain results near phi. Same holds for other body parts. Surprisingly the better the approximation is, the more well proportioned the individual is usually considered. That's why ancient sculptures follow that principle.

The golden proportion is approximated by the ratios of adjacent terms of the sequence

1,1,2,3,5,8,13,21,34,55,89,144... (called Fibonacci's series)

each term of which is given by adding the two terms before it. You are invited to convince yourself, that:

1/1=1

2/1=2

3/2=1.5

5/3=1.66...

8/5=1.6

13/8=1.625

21/13=1.615384615384...

34/21=1.61904761904761904...

55/34=1.61764705882352941176470588235294...

You see, that the ratios approximate phi. A proof of this should be found in most lectures about analysis held in the first semester. I don't want to write it down without knowing if you want one. Fibonacci allegedly discovered it by monthly counting rabbit pairs in a rabbit population (not a joke, but yet an other example of phi's everlasting presence!)

Enjoy the study of this ancient and astonishing ratio! I wish you success, hopeful to have both helped you and made you curious about this great section!!

Number Theory > Constants > Golden Ratio

Number Theory > Constants > Algebraic Constants

Number Theory > Constants > Continued Fraction Constants

History and Terminology > Notation

History and Terminology > Terminology

Recreational Mathematics > Mathematics in the Arts > Mathematical Sculpture

Recreational Mathematics > Mathematics in the Arts > Mathematics in Literature > The Da Vinci Code

Recreational Mathematics > Mathematics in the Arts > Mathematics in Television > NUMB3RS

MathWorld Contributors > Cloitre

MathWorld Contributors > Cook, Richard

MathWorld Contributors > Lambrou

Golden Ratio

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

The Da Vinci Code book has quite and explaination on the Golden Ratio. This ratio is constant in the bigger numbers of the Fibonacci Sequence. You can get a lot of help from the book and can add its name in the Bibliography.

Its value is 1.618.....

It determines how likely a guy is to touch himself sensually when having an involountary Golden Shower (i.e. rain of piss).

One example is Pi.U can go Wikipedia to search info on it.

it is the ratio of a particular rectangle

check out the following its informative

http://goldennumber.net/

hope this helps:

http://en.wikipedia.org/wiki/Golden_ratio