Science&Mathematics ④ bEttEr fUtUre: the Golden Ratio

the Golden Ratio

Posted by ^seng^ , Labels: mysterious of world , Monday, March 15, 2010 8:16 PM

Golden Ratio & Golden Section : : Golden Rectangle : : Golden Spiral

Golden Ratio & Golden Section

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Expressed algebraically:

The golden ratio is often denoted by the Greek letter phi (Φ or φ).

The figure of a golden section illustrates the geometric relationship that defines this constant. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Golden Rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: j (one-to-phi), that is, 1 : or approximately 1:1.618.

A golden rectangle can be constructed with only straightedge and compass by this technique:

Construct a simple square
Draw a line from the midpoint of one side of the square to an opposite corner
Use that line as the radius to draw an arc that defines the height of the rectangle
Complete the golden rectangle

Golden Spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Specifically, a golden spiral gets wider (or further from its origin) by a factor of j for every quarter turn it makes.

Successive points dividing a golden rectangle into squares lie on

a logarithmic spiral which is sometimes known as the golden spiral.

Image Source: http://mathworld.wolfram.com/GoldenRatio.html

Golden Ratio in Architecture and Art

Many architects and artists have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. [Source: Wikipedia.org]

Here are few examples:

Parthenon, Acropolis, Athens.

This ancient temple fits almost precisely into a golden rectangle.

Source: http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm

The Vetruvian Man"(The Man in Action)" by Leonardo Da Vinci

We can draw many lines of the rectangles into this figure.

Then, there are three distinct sets of Golden Rectangles:

Each one set for the head area, the torso, and the legs.

Leonardo's Vetruvian Man is sometimes confused with principles of "golden rectangle", however that is not the case. The construction of Vetruvian Man is based on drawing a circle with its diameter equal to diagonal of the square, moving it up so it would touch the base of the square and drawing the final circle between the base of the square and the mid-point between square's center and center of the moved circle:

Golden Ratio in Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.[38] Zeising wrote in 1854:

The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.

Examples: