Golden Ratio

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Golden ratio The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid,[1] the Parthenon, the Colosseum. The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclid in the 5th century BC. In the fourth century BC, Aristotle noted its aesthetic properties. [3] Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Keplers triangle, were believed to be aesthetically pleasing. As such, many works of ancient art exhibit and incorporate the golden ratio in their design. Summerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from as early as the late Bronze Age.[The prevalence of this special number in art and architecture even before its formal discovery by Pythagoras is perhaps evidence of an instinctive and primal human cognitive preference for the golden ratio. [5] Golden ratio in Pyramids Evidence of mathematical influences in art is present in the Great Pyramids, built by Egyptian Pharaoh Khufu and completed in 2560BC. Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments. They note that the length of the base edges range from 755756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet. [6] If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. This would also indicate that half the cross-section of the Khufus pyramid is in fact a Keplers triangle Egyptians told that the dimensions of the Great Pyramid were so chosen that the area of a square whose side was the height of the great pyramid equaled the area of the triangle and if this is true than it would undeniably prove the intentional presence of the golden ratio in the pyramids Golden Rectangle A golden rectangle is one whose side lengths are in the golden ratio, approximately 1:1.618. or

A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989 Other names frequently used for the golden ratio are the golden section and golden mean The golden ratio is denoted by the Greek lowercase letter phi (), while its reciprocal, 1 / or 1, is denoted by the uppercase variant Phi ().

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has one positive solution in the set of algebraic irrational numbers:

.

Let smaller part = 1, larger part = . Thus is the golden ratio. It is often designated by the greek letter phi, for Phideas, (fl. c. 490-430 BC), Athenian sculptor and artistic director of the construction of the Parthenon, who supposedly used the golden ratio in his work. Then by the definition of the golden ratio, / 1 = (1 + so2

)/

= 12 + 1

and we get the quadratic equation,2

- -1=0

As a project, solve this quadratic equation for the golden ratio . You should get, = 1/2 + 5/2 1.618

Given a rectangle having sides in the ratio , the golden 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. Euclid used the following construction to construct them. Draw the square , call the midpoint of so that . Now draw the segment , which has length

,

(1)

and construct

with this length. Now complete the rectangle

, which is golden since