The Golden Ratio

The Golden Ratio

In mathematics, painting, architecture, book design, music, nature and more, many have proportioned their works to approximate the golden ratio. Many believe this proportion to be aesthetically pleasing.

Mathematicians have studied the golden ratio because of its unique and interesting properties. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. The golden ratio is often denoted by the Greek letter Φ (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

The golden section is a line segment sectioned into two according to the golden ratio. The total length a + b is to the longer segment a as a is to the shorter segment b.

Timeline

• Phidias (490–430 BCE) made the Parthenon statues that seem to embody the golden ratio.
• Plato (427–347 BCE), in his Timaeus, describes five possible regular solids (the Platonic solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron), some of which are related to the golden ratio.
• Euclid (c. 325–c. 265 BCE), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio"
• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the Fibonacci sequence is closely related to the golden ratio.
• Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
• Johannes Kepler (1571–1630) describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle.
• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
• Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.
• Edouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
• Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.
• Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.