Golden Section

To understand what constitutes a golden section, consider the top line segment shown on the next page. The line segment has three of its points marked. Point P partitions the line segment AB into two smaller segments: from left endpoint A to point P (AP ), and from P to right endpoint B (PB ). The line segments AP and PB are also shown individually.

A "golden ratio" can now be formed as the ratio of one line segment to another line segment. Point P divides the entire line segment AB into a golden section if the following equation is valid:

AP/AB = PB/AP.

In other words, the length of line segment AP divided by the length of the entire line segment AB is a ratio, or number, and that number is the same as the ratio of the shortest line segment PB and the segment AP. It turns out that these ratios are equal to the irrational number 0.61803…; that is:

AP/AB = PB/AP = 0.61803 …

Note that the decimal places continue indefinitely. The number 0.61803…is called the golden ratio or golden number, whereas the term "golden section" refers to the line segments formed by a point, such as P, whose resulting ratio is the golden number.

Significance of the Golden Ratio

The golden section and the golden ratio are important to mathematics, art, and architecture. The construction of the golden section goes back at least as far as the ancient Greeks, to the Pythagoreans (ca. 550 b.c.e.–300 b.c.e.). The Pythagoreans believed that numbers were the basis of nature and man, and therefore number relationships like the golden ratio were of utmost importance.

Besides line segments, the golden ratio also appears in many geometric figures. For example, a rectangle is said to be in golden section if its width and length are in the golden ratio (that is, if the width divided by the length equals 0.61803…).

Some scholars believe that various temples of the ancient Greeks, like the Parthenon in Athens, were purposefully produced with various dimensions in the golden ratio. Many of the artists and architects of the Renaissance period are believed to have incorporated the golden ratio into their paintings, sculptures, and monuments. A prime example is Leonardo da Vinci (1452–1519), who extensively used golden ratios (or their approximations) in his paintings. In Leonardo's famous drawing "Vitruvian Man," the distance ratio from the top of the head to navel, and from the navel to the soles of his feet approximates the golden ratio.

Many people feel that geometric forms and figures incorporating the golden ratio are more beautiful than other forms. Psychological studies purportedly show that people find golden-ratio rectangles more appealing than rectangles of other proportions.

see also Architecture; Leonardo da Vinci; Numbers, Forbidden and Superstitious; Pythagoras.

Philip Edward Koth with

William Arthur Atkins

Bibliography

Eves, Howard. An Introduction to the History of Mathematics, 4^th ed. New York: Holt, Rinehart and Winston, 1976.

lnternet Resources

The Golden Section Ratio: Phi. Department of Computing, University of Surrey, United Kingdom. <http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html>.

ALTERNATIVE DEFINITION OF GOLDEN SECTION

Some dictionaries and textbooks define golden section (and number) as the inverse of the definition shown in this article. The formula then becomes (again referring to line segment AB ) AB /AP = AP /PB = 1.61803….

Sometimes the larger value is denoted by the Greek letter "Phi"(i.e., Phi = 1.61803…) while the smaller value of the golden number is denoted by a "small" phi (i.e., phi = 0.61803…). Note that Phi 1 + phi.

golden section. Also called the golden cut or mean, or harmonic proportional ratio, it may have originated in C6 bc in the circle of Pythagoras, was certainly known during the time of Euclid (c.325–c.250 bc), and was held to be divine by several Renaissance theorists, especially Luca Pacioli (c.1445–c.1514) in his De Divina Proportione, written in 1497 and published in Venice in 1509. It can be expressed as a a straight line (or as a rectangle) divided into into two parts so that the ratio of the shorter part (a) to the longer (b) is the same as the ratio of the longer (b) to the sum of the shorter and longer parts, or a:b = b:a + b, or that the ratio of the smaller part is to the longer as the latter is to the whole. The ratio is expressed in algebra as Φ (Phi, the first letter of the name of the Greek sculptor Phidias, or Pheidias (c.490–430 bc) ) = (1 +√5)/2, which comes to 1.61803. Thus the ratio is approximately 8: 13.

Bibliography

Borrisavlievitch (1970);
Ghyka (1976);
Hagenmaier (1977);
Huntley (1970)