There is an ancient number that has memorized scholars, artists, scientists, architects, musicians, and mathematicians for ages. This number is believed to be the equation for things perceived as being aesthetically pleasing. This mysterious number is the Golden Ratio which equals approximately 1.61803 (Walser, 4).

Represented by the Greek lowercase letter phi (φ), also known as the Golden Ratio, this is by far one of the most astonishing numbers known to man. It can also be referred to more commonly as the golden section, but also the golden mean, divine proportion, divine section, golden proportion, golden number, and medial section, amongst other names given throughout the ages (Golden Ratio).

The characteristics of “the Golden Ratio is thus the ratio of the larger subsegment to the smaller” (Walser). The proportioning of the golden ratio, drawn into the golden spiral, is believed to be aesthetically pleasing, thus resulting in the belief that this could be the mathematical equation for beauty. Artists proportion their works in accordance to the golden ratio. Architects have been designing structures, such as the pyramids, the Greek’s heavily used this ratio, and it commonly occurs in nature.

The golden ratio can be considered math in art and can be mathematically expressed using the Golden Section as: a + b / a = a / b = φ.  That, in turn, yields the solution: φ = 1+ √5 / 2 = 1.6180339887. The golden section is a line segment, divided into two sections where the whole line segment equals a + b, but the segment is divided into a and b, where a is the longer segment and b is the shorter. The reason this number is so interesting is due to the fact that it is an irrational constant with positive properties. (Golden Ratio)
Ancient Greek’s were the first to study this, which was then known as Pythagoreans symbol. This is where the Golden Section was given its title due to the “extreme and mean ratio” because of its importance to the geometry of regular pentagrams and pentagons (Wikipedia). However, “Euclid's Elements provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."” (Golden Ratio).

The golden ratio has been represented by the Greek lowercase letter phi, since the twentieth century after a sculptor by the name of Phidias (phi). This Greek letter has become famous and heavily attached to representing the golden ratio. Phidias is believed to be one of the greatest sculptors of Classical Greek. He sculpted many of the Greek Gods, as to which he became immortalized for. (Phidias)

During the Renaissance, large amount of ideas emerged on the aesthetics of the golden ratio were written and developed. De Divina Proportione by Luca Pacioli explored the proportions and mathematics of how this golden ratio is related to art and artistic proportions. It is widely noted and believed that Leonardo Da Vinci’s Vitruvian Man was proportioned according to this golden ratio. Many scholars believe that Da Vinci may have been heavily influenced by the golden ratio, due to the way he proportioned his works, despite the fact that he has not written of its employment in his works. On the other hand, Salvador Dali uses this famous ratio in “The Sacrament of the Last Supper,” as to which by simply viewing the painting you can note the proportions held within the painting and notice how eye catching it is. This beauty could only be obtained by such a ratio as the golden ratio. (Golden Ratio) (Walser)

Commonly found in architecture is the golden ratio and the golden rectangle. Despite the debate amongst scholars, it has been reviewed and believed by most, that the proportions of this ratio holds true in many artistic designs across the field. To demonstrate how important “Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture” (Golden Ratio).

There have been many composers and musicians who followed the golden ratio. It has been observed that Debussy followed this ratio in his compositions, along with such note worthy composers such as Chopin, amongst others. Of course these are simply theories developed through the research and observation of musicologists and not the artists themselves. (Golden Ratio)

An interesting observation was made about the relationship between nature and the golden ratio as stated by “Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law” (Golden Ratio).

Within mathematics, the golden triangle, more commonly known as an isosceles triangle with a bisecting line to divide it proportionally to the golden ratio. The golden ratio can be proven and functional in a wide variety of geometry and algebra. This can be confirmed by Ptolemy’s  theorem b² = a² +ab, which gives the answer of b/a = (1+ √5) / 2 or φ. (Golden Ratio) (Walser)

More note worthy than anything else is the golden ratios relationship to Fibonacci’s sequence. Fibonacci’s sequence begins with the numbers 0 and 1 and each number from that point on is the sum of the previous. The sequence is commonly taught in early math classes and follows the format of 0,1,1,2,3,5,8,13,21,34,55, and so on. Fibonacci’s sequence brings a new light towards the golden spiral, which is equal to Fibonacci’s spiral. The formula for the Fionacci numbers equals phi and can be proven mathematically. Fibonacci’s numbers are the most significant relationship the golden ratio has to anything mathematically. The golden spirals can be found in storm patterns, the blooming patterns of flowers, sea shells, amongst other natural occurrences. It has been noted that the golden ratio weighs heavily in Fibonacci’s numbers and can be proven that “if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ” (Golden Ratio). (Golden Ratio) (Walser)

From sculptures, paintings, sketches, buildings, monuments, to mathematics- everything can be connected to this magnificent ratio. Leonardo Di Vinci found importance in it, as seen evident in his paintings. Composers produced their music to the length and timing of this Ratio or Section. Nature grows in accordance to this. This is a universal equation to help describe, mathematically, why we like the things we do.

Works Cited

Wikipedia contributors. "Golden ratio." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 29 Mar. 2010. Web. 3 Apr. 2010. http://en.wikipedia.org/wiki/Golden_ratio (Golden Ratio)

Wikipedia contributors. "Phidias." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 19 Mar. 2010. Web. 3 Apr. 2010. http://en.wikipedia.org/wiki/Phidias (Phidias)

Walser, Hans, British Division, British Services, and Council Bibliography. British national bibliography. The Mathematical Association of America, 2001. Print. (Walser)

### Quick Facts

Nationality - Italian
Place of Birth - Vinci, Italy
Born - April 15, 1452
Died - May 2, 1519
Education - Apprentice to Andrea del Verrocchio
Patrons - Pope Leo X, Ludovico Sforza the Duke of Milan, Cesare Borgia and King Francis I of France