Golden Ratio Calculator

The golden ratio, written as phi, equals (1 plus the square root of 5) divided by 2, or 1.6180339887 to ten decimal places. It is the positive root of x squared minus x minus 1 equals zero, which gives it two properties that make the multiply and divide operation on this page meaningful: phi squared equals phi plus 1, and 1 divided by phi equals phi minus 1, or 0.6180339887. So for an input of 100, the two partners are 161.80 and 61.80, and those three numbers form a chain in which each term is the sum of the two below it, since 61.80 plus 100 equals 161.80. The ratio is tied to the Fibonacci sequence because consecutive terms converge on it from alternating sides: 13 divided by 8 gives 1.625, 21 divided by 13 gives 1.6154 and 89 divided by 55 gives 1.61818. Because phi has the continued fraction expansion consisting entirely of ones, it is the real number least well approximated by rationals, which is the reason plants using the derived golden angle of about 137.5 degrees pack successive leaves or seed primordia without ever aligning into wasteful rows. That botanical case is genuine and measurable; many of the more famous claims are not. Proportions read off the Parthenon, the Great Pyramid or the Mona Lisa depend heavily on which edges the measurer chooses, and the nautilus shell is a logarithmic spiral with a growth factor closer to 1.33 than to phi.

Euclid gave the first surviving rigorous treatment around 300 BCE, defining in Book VI of the Elements what he called cutting a line in extreme and mean ratio. Leonardo of Pisa, known as Fibonacci, brought the additive sequence to European readers in Liber Abaci in 1202, without connecting it to Euclid's ratio. Johannes Kepler made that link explicit in the early seventeenth century, noting that ratios of successive terms in the sequence approach the divine proportion. Luca Pacioli supplied the name that stuck with De divina proportione, printed in Venice in 1509 and illustrated by Leonardo da Vinci, and the symbol phi came into use through Mark Barr around 1909 in honour of the sculptor Phidias. A closed form expressing the nth Fibonacci number in terms of powers of phi was published by Jacques Philippe Marie Binet in 1843, though Abraham de Moivre had derived it by 1730. The aesthetic claim came separately and later, from Gustav Fechner, who reported in 1876 that subjects asked to pick a preferred rectangle clustered near the golden shape; subsequent replications have not established a reliable effect. The proportion acquired genuine physical significance in the late twentieth century when Roger Penrose described aperiodic tilings in 1974 whose two rhombs relate by phi, and Dan Shechtman discovered quasicrystals in 1982, whose icosahedral diffraction patterns show phi-scaled spacings. Mario Livio's The Golden Ratio in 2002 separated these documented cases from the accumulated folklore.