Golden Section


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Golden Section,

in mathematics, division of a line segment into two segments such that the ratio of the original segment to the larger division is equal to the ratio of the larger division to the smaller division. If c is the original segment, b is the larger division, and a is the smaller division, then c = a + b and c/b = b/a. Thus, b is the geometric mean of a and c; the ratio is known as the Divine Proportion. The Golden Rectangle, whose length and width are the segments of a line divided according to the Golden Section, occupies an important position in painting, sculpture, and architecture, because its proportions have long been considered the most attractive to the eye. The constructions of regular polygons of 5, 10, and 15 sides depend on the division of a line by the Golden Section. The numerical ratio of the greater segment of the line to the shorter segment as determined by the Golden Section is symbolized by the Greek letter phi and has the approximate value 1.618. It occurs in many widely varying areas of mathematics. For example, in the Fibonacci sequence (the sequence of numbers formed by adding successive members to find the next member—0, 1, 1, 2, 3, 5, 8, 13, … ), the values of the ratios 1, 2/1, 3/2, 5/3, 8/5, 13/8, … approach the value of the Golden Section.

Bibliography

See H. E. Huntley, The Divine Proportion (1970).

Golden Section

 

the division of a line segment AB into two parts in such a way that its larger part AC is the mean proportional between the entire segment AB and its smaller part CB (see Figure 1).

Figure 1

The algebraic determination of the golden section of the segment AB = a reduces to solving the equation a/x = x/(ax) (where x = AC), from which x = a ( Golden Section − 1)/2 ≈ 0.62a. The ratio of x to a may also be expressed approximately by the fractions ⅔, ⅗, ⅝, 8/13, 13/21, and so on, where 2, 3, 5, 8, 13, 21, and so forth are Fibonacci numbers. The geometric construction of the golden section of the segment AB is accomplished in the following manner: a perpendicular to AB is drawn at point B, the segment BE = 1/2AB is measured off, A and E are connected, ED = EB is measured off, and finally, AC = AD, thus AB/AC = AC/CB. The golden section was already known in antiquity. In the classical literature that has come down to us, the golden section is first encountered in Euclid’s Elements (third century B.C.). The term “golden section” was introduced by Leonardo da Vinci (the turn of the 16th century). The principles of the golden section and the proportional relationships close to it have served as the basis for the compositional construction of many works of world art (chiefly architectural works of ancient Greece and Rome and the Renaissance).

golden section

[′gōl·dən ′sek·shən]
(mathematics)
The division of a line so that the ratio of the whole line to the larger interval equals the ratio of the larger interval to the smaller. Also known as divine proportion; extreme and mean ratio; golden mean; golden ratio.

golden section

The division of a line into two segments so that the ratio of the whole line to the larger segment equals the ratio of larger segment to the smaller one; often called the golden section. This ratio was once considered by some as having an inherent aesthetic value.
References in periodicals archive ?
At the moment we can only state that main maxima in this distribution must corresponds to co-called golden section (10).
The golden section method requires no information about the derivative of the function.
He tells us that squaring the circle and the Golden Section are fascinating topics in mathematics, influential in art and architecture down to the present day, and we believe that too.
And so, it is no wonder that Georgian architects, designers and craftsmen, aware of the system of rules and proportion of the Orders of Architecture and the ratio of the Golden Section, were well-equipped to create beauty in everything they made.
Included in the PBS broadcast will be Western Symphony and another Tharp ballet, the high-flying Golden Section.
Math for mystics; from the Fibonacci sequence to Luna's Labyrinth to the Golden Section and other secrets of sacred geometry.
Led by artistic director Judith Jamison, AAADT will be performing three works in Newcastle - Firebird, The Golden Section and the company's signature piece Revelations.
Firebird is a new take on the Stravinsky ballet, The Golden Section sees dancers frantically filling the floor space to tunes by David Byrne and Revelations is a series of scenes set to African-American religious music.
The architect availed himself of the opportunity to explain his work on the Modulor system, an attempt to locate the golden section proportional to the height of the average person.
In addition to Alvin Ailey's beloved Revelations, Alvin Ailey American Dance Theater's 2007 Atlanta engagement will feature the highly anticipated company premiere of The Golden Section by Tony Award-winning choreographer Twyla Tharp with music by David Byrne and newly-designed costumes by Santo Loquasto.
In his case, the common feature was a numerical order based on the figure seven and a geometrical order based on the Golden Section, playing upon its property of squares interlocking spirally seven times (seven mortal sins, seven cardinal virtues, seven days of Dante's tour on 7 April 1300