Golden Section

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. A proportion in which a straight line or rectangle is divided into two unequal parts in such a way that the ratio of the smaller to the greater part is the same as that of the greater to the whole. Like the mathematical value pi, it cannot be expressed as a finite number, but an approximation is 8:13 or 0.618:1. The proportion has been known since antiquity (it is discussed by Euclid and Vitruvius) and has been said to possess inherent aesthetic value because of an alleged correspondence with the laws of nature or the universe. It was much studied during the Renaissance, and Luca Pacioli (c.1445–c.1514), the most famous mathematician of his day and a friend of Leonardo and of Piero della Francesca, wrote a book on it called Divina proportione (1509); some of its illustrations are by Leonardo. In accordance with the tendencies of the time, Pacioli credits this ‘divine proportion’ with various mystical properties and exceptional beauties both in science and in art. Like many other learned men of the Middle Ages and Renaissance, he was anxious to harmonize the knowledge of pagan antiquity with the Christian faith, and in the chapter in which he justifies his choice of title he explains that this ratio cannot be expressed by a number and, being beyond definition, is in this respect like God, ‘occult and secret’; further, this three-in-one proportion is symbolic of the Holy Trinity.

golden section A proportion in which a straight line or rectangle is divided into two unequal parts in such a way that the ratio of the smaller to the greater part is the same as that of the greater to the whole. Like the mathematical value pi, it cannot be expressed as a finite number, but an approximation is 8 : 13 or 0.618 : 1. The proportion has been known since antiquity and has been said to possess inherent aesthetic value because of an alleged correspondence with the laws of nature or the universe. It was much studied during the Renaissance, and Luca Pacioli, the most famous mathematician of his day and a friend of Leonardo and of Piero della Francesca, wrote a book on it called Divina proportione (1509). In accordance with the tendencies of the time, Pacioli's book, illustrated with drawings by Leonardo, credits this ‘divine proportion’ with various mystical properties and exceptional beauties both in science and in art.