In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. It is an irrational number with a value of 1.6180339887. The Greek letter phi represents the golden ratio and is the limiting ratio of consecutive Fibonacci numbers.
The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence. According to Mario Livio. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.
Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". Fibonacci popularized the Hindu–Arabic numeral system in the Western World primarily through his composition in 1202 of Liber Abaci. He also introduced Europe to the sequence of Fibonacci numbers, The ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.
Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion". De Divina Proportione, a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.
Leonardo da Vinci's illustrations of polyhedra in De divina proportione and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings. Similarly, although the Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.
The golden ratio has persistently been claimed in modern times to have been used in art and architecture by the ancients in Egypt, Greece and elsewhere. Also in modern times the divine proportion have continue been used by artist like Salvador Dalí and a variety of modern architects.
The Renaissance Academy is looking to continue exploring the golden ratio in different art forms.